Momentum Model-based Minimal Parameter Identification of a Space Robot
aa r X i v : . [ c s . R O ] S e p Momentum Model-based Minimal Parameter Identificationof a Space Robot
B. Naveen ∗ and Suril V. Shah † Indian Institute of Technology, Jodhpur, Rajasthan, India, 342011
Arun K. Misra ‡ McGill University, Montreal, Quebec H3A 0C3, Canada
Accurate information of inertial parameters is critical to motion planning and control ofspace robots. Before the launch, only a rudimentary estimate of the inertial parameters isavailable from experiments and computer-aided design (CAD) models. After the launch, on-orbit operations substantially alter the value of inertial parameters. In this work, we propose anew momentum model-based method for identifying the minimal parameters of a space robotwhile on orbit. Minimal parameters are combinations of the inertial parameters of the linksand uniquely define the momentum and dynamic models. Consequently, they are sufficientfor motion planning and control of both the satellite and robotic arms mounted on it. Thekey to the proposed framework is the unique formulation of momentum model in the linearform of minimal parameters. Further, to estimate the minimal parameters, we propose anovel joint trajectory planning and optimization technique based on direction combinations ofjoints’ velocity. The efficacy of the identification framework is demonstrated on a 12 degrees-of-freedom, spatial, dual-arm space robot. The methodology is developed for tree-type spacerobots, requires just the pose and twist data, and scalable with increasing number of joints.
Nomenclature n = Number of 1-DoF joints in the space robot L i = Link with index iJ i = Joint with index i Σ I = Inertial frame of reference Σ i = Reference frame attached to L i based on modified Denavit–Hartenberg (DH) notation Σ ( c ) i = Reference frame attached to L i at its center of mass having the same orientation as Σ i c i ∈ R = Position vector from Σ I to Σ ( c ) i ∗ Research Assistant, Department of Mechanical Engineering, N.H. 65 Nagour Road; [email protected] † Assistant Professor, Department of Mechanical Engineering, N.H. 65 Nagour Road; [email protected] ‡ ThomasWorkman Professor, Department of Mechanical Engineering, 817 Sherbrooke Street West; [email protected]. Fellow AIAA i ∈ R = Position vector from Σ I to Σ i a i ∈ R = Position vector from Σ i to Σ ( c ) i v i , ω i ∈ R = Linear velocity and angular velocity of Σ i p i , l i ∈ R = Linear and angular momentum of L i respectively p , l ∈ R = System linear and angular momentum respectively K i ∈ R × = Link kinematic matrix of L i G ∈ R × ( n + ) = Global kinematic matrix i R j ∈ SO(3) = Rotation matrix from Σ i to Σ j R i ∈ SO(3) = Rotation matrix from Σ I to Σ j m i ∈ R = Mass of L ii a i ∈ R = Position vector from Σ i to Σ ( c ) i expressed in Σ ii b j ∈ R = Position vector from Σ i to Σ j expressed in Σ i ρ j ∈ I + Ð { } = Parent link index of j th link α j = Joint twist between Σ ρ j and Σ ji I ( c ) i ∈ R × = Inertia matrix of L i around Σ ( c ) ii I i ∈ R × = Inertia matrix of L i around Σ i , O , E = Zero vector, Zero matrix, and Identity matrix respectively of compatible dimensions v ( j ) = j th component of vector v φφφ i ∈ R × = Link parameter vector of L i φφφ ∈ R ( n + )× = Standard parameter vector t i = i th time instant θ js , θ jd ∈ R = Start and range parameters of the j th joint w ∈ R + = Weighing factor κ (·) = Condition number N t = Number of trajectories of rates (Six for base twist + n for joints rates) N m = Number of instants of discretization of the trajectory h ij = Base/joint velocity from the i th trajectory at j th instant θ kji , θ kj f = Interval trajectory parameters, initial and final joint position respectively of the j th joint in the k th interval t = Time measured since the beginning of the k th interval T p = Time period of the interval q j = Joint position ǫ (·) = Root mean squared (RMS) error 2 j , , q ( s ) j , i and q ( c ) j , i = Coefficients of fourier series for the j th joint N h = Number of harmonics in the fourier series ω f = Angular frequency of the fourier seriesAll the position and velocity vectors are expressed in Σ I unless mentioned. I. Introduction O ver the past two decades, space robots have gained prominence by performing on-orbit tasks, such as assembly,inspection, refueling, and docking [1]. They are also a potential solution to remove space debris and harvestretired satellites [1]. On-orbit tasks demand an accurate motion planning and control of a space robot, which in turn relyon the kinematic and dynamic models [2–5]. Unlike a fixed-base robot, the kinematics of a space robot is a functionof the mass, the position of center of mass (CoM), and the inertia tensor of the links, evident from the GeneralizedJacobian Matrix [2]. Consequently, to perform kinematic and dynamic analysis, motion planning, and control, theknowledge of inertial parameters is indispensable.For a rigid link, there are ten inertial parameters, i.e., one with a mass, three with a position of CoM, and six withan inertia tensor. All of them can be identified for individual links before the system assembly [6, 7]. However, itdemands a complicated experimental setup. Further, in a dismantled state, the links do not include cables for powertransfer and communication and piping for fluid supply, which are quite heavy [8]. Also, the CAD models provideestimates of inertial parameters which are erroneous by more than 10% [7]. Moreover, an inaccurate estimate of theinertial parameters adversely affects the motion planning and control [9]. Furthermore, while operating on-orbit, achange in the fuel quantity and payload can substantially alter the inertial parameters [10, 11]. Neither the pre-assemblyexperiments nor the CAD models account for such an on-orbit change in the inertial parameters. A solution to addressall the above-mentioned issues is a model-based identification of the inertial parameters of the robot while on orbit.In this work, we focus on identifying the minimal parameters of a space robot based on the momentum modelwhile on orbit. The minimal parameters define the dynamic model uniquely [12, 13]. They are a linear combinationof the link parameters, which are the mass ( m i ), the product of mass and position of CoM ( m ii a i ) components, and theinertia tensor (cid:0) i I i (cid:1) components of multiple links. Such a linear combination is due to constraining the relative motionof an adjacent pair of links with a joint [12], referred as kinematic constraints. The inertial parameters are redundantin defining the dynamic model; consequently, only the minimal parameters can be identified [14]. Hence, our focus ison identifying the minimal parameters; however, by using the momentum model.The central proposition of the presented work is to formulate the momentum model in the linear form of minimalparameters. The proposition is built on the evidence that the dynamic model can be formulated in the linear form ofminimal parameters [13–15]. Formulating the momentum model in terms of minimal parameters makes them sufficient3ot only for momentum model-based control but also for motion planning and dynamic model-based control. Estimatingsuch widely applicable minimal parameters of the space robot, with no prior knowledge of inertial parameters of anyof its links, fundamentally distinguishes the presented work from other momentum-based identification techniques[10, 16–19].Momentum model defining the kinematics is characteristic of a space robot. Using such a characteristic featurefor minimal parameter identification, based on the proposed linear formulation of momentum model, has the followingadvantages: • Minimal parameters fully construct the momentum and dynamic models. Hence, they are sufficient for modeling,motion planning, and control. • Momentum model consumes low noise data; thus, results in accurate estimates. • Identification framework does not require thrusters and fuel resources. Instead, electrically driven actuators areused. • Momentum model is computationally efficient in comparison to dynamic model to use with adaptive controltechniques under no external force and torque. • Linear formulation is also the key to design computationally efficient trajectories for identifying the minimalparameters.Firstly, a momentum model requires only pose and twist while a dynamic model needs torque and acceleration aswell. On the one hand, position and velocity are easy to measure, and their measurements have high signal-to-noise(SN) ratio. On the other hand, torque and acceleration measurements often have low SN ratio in space robots since theirsignal amplitude is low for safety reasons [16, 17]. As a result, parameter estimates based on the momentum modelare proven to be substantially accurate than dynamic model based parameter estimates [10, 18]. Next, the proposedframework uses only robotic arms and reaction wheels for parameter identification. They consume only electricity;thus, preserving the limited fuel resources. Finally, an exact linear model is a significant component of an adaptivecontroller. Momentum model has only six equations for a system with an arbitrary number of links and consumes loweramount of data than the dynamic model. Thus, it is computationally efficient for designing an adaptive controller.
Contributions and Paper Organization.
The primary contribution of this work is a systematic procedure for linearformulation of the momentum model in terms of the minimal parameters. In the usual momentum formulation of aspace robot [20, 21], inertial parameters are indistinct because they are grouped with kinematic data (base pose andtwist, and joint angle and velocity) and geometric parameters (link length, joint twist, and joint offset). However, inSec. III, the inertial parameters are systematically separated from the kinematic data and grouped into link parameters.Further, we obtain the System Kinematic Matrix (SKM) and standard parameters for a generic space robot. SKMcontains the kinematic data of all the links of the system and links parameters of all the links are referred to as standardparameters. SKM linearly transforms the standard parameter vector to system momentum.4n Sec. IV, the kinematic constraints due to the joints are imposed onto the momentum model. The kinematicconstraints regroup the standard parameters resulting in a momentum model in the linear form of minimal parameters.A recursive approach is provided to obtain the closed-form solution of minimal parameters for a generic tree-type spacerobot based on only the geometric parameters of the system.In Sec. V, computation of exciting joint trajectories is presented. They render all the minimal parameters identifiableand facilitate in estimating the parameters which are minimally sensitive to noise. The whole system usually cannot beexcited with a single motion [22], and it presents a major challenge in computing exciting trajectories. It is tackled witha novel joint trajectory planning and optimization technique for computing exciting trajectories, which forms anothercontribution of the work. The novelty lies in composing the joint trajectory with multiple intervals to excite all theminimal parameters of the system. Each interval adds new information about the system behavior while improving thenumber of identifiable minimal parameters. Moreover, the proposed approach is computationally efficient and scalablewith increasing degrees-of-freedom (DoF).Finally, in Sec. VI, we apply the identification procedure on a 12-DoF, spatial, dual-arm space robot and estimate itsminimal parameters with measurement noise in the kinematic data. Further, the performance of the estimated minimalparameters is evaluated by predicting the system’s kinematics and dynamics. The proposed method uses reactionwheels with known inertial parameters to apply momentum onto the robot. We assume that the minimal parameters ofthe space robot do not change while executing the exciting trajectories. Further, the geometric parameters are usuallyprecisely known [13, 17] and assumed to be deterministic in nature.
II. Related work
Solving the problem of parameter identification usually entails model formulation,experiment design, and parameterestimation. We discuss the related literature from the perspective of these tasks.
A. Modeling
Identification techniques based on both momentum [10, 16–19] and dynamic [8, 10, 11, 14, 23–25] models exist.Many works involved estimating the parameters of one of the links, with the knowledge of inertial parameters of restof them [10, 11, 17, 19, 24]. However, frameworks which identify atleast some inertial parameters or a combination ofinertial parameters of all the links is referred in this work as whole-system identification.In one of the initial attempts towards whole-system identification, angular momentum model was linearized aroundmass and inertia parameters using partial differentiation [16]. Adapting such a linearization to all the inertial parametersprovides only an approximate linear model because of nonlinearity in the CoM parameters. Moreover, the frameworkin [16] estimates only some of the parameters of all the links while needing the knowledge of others.Recently, a fusion of dynamic and momentum model was performed to estimate the inertial parameters of all the5inks [20]. However, the procedure needs firing the thrusters and requires high-noise acceleration measurements. Acombination of inertial parameters of the space robot are estimated using only the angular momentum model in [18]. Itis the work that comes closest to whole-system identification based only on momentum model. However, the estimatedparameters are limited to model-based control of attitude only. Hence, problems related to the position and linearvelocity control, motion planning and collision avoidance in task space cannot be solved.In [22], minimal parameters were estimated by considering only the base link’s dynamic model. Thus the needfor joint torque and joint acceleration are eliminated. Even though the methodology applies to generic tree-typefloating-base systems, it is suitable for systems like humanoids and humans. They have huge acceleration leading tohigher SN ratio in their base link’s acceleration measurements as compared to space robots. The base link is a satellitein space robots, which has a very low acceleration for safety reasons [17].It is evident that the whole-system identification involving dynamic models require high-noise measurements andmay need fuel resources. Using the momentum model only some parameters are estimated, which are insufficient formotion planning and control in task space.
B. Experiment Design
The designed trajectories which facilitate identifying the parameters of interest are usually called exciting trajectories.An approach widely used for fixed-base robots is adapted for space robots in [18, 19, 24] by minimizing the conditionnumber of the regressor matrix. A well-conditioned regressor matrix minimizes the effect of noise on the parameterestimates [26]. Also, the experiment design has to ensure that the SN ratio of the measured data is high [11] so thatthe perturbation in its ground truth is minimum. Both condition number and SN ratio are utilized in the current work.In [24], using ideas from optimal control, exciting trajectories were designed within 8 minutes for a single link6-DoF system. However, with increasing number of dimensions, optimal control strategies often consume hours ofcomputation time [5]. Furthermore, exciting all the minimal parameters with a single motion is difficult [22]. Thedifficulty increases with increasing DoF of the robot. This problem was partially solved by running a few intuitivelyselected trajectories on the robot [14]. We identify that designing a systematic methodology for computing excitingtrajectories as another area for research and propose a solution to it in this work. In Table 1, a comparison of variouswhole-system identification techniques is shown.
C. Estimation Methodology
In [20], owing to an unprocessed form of the momentum model, which is nonlinear in terms of parameters,particle swarm optimization was implemented for parameter estimation. Formulations based on a processed form ofmomentum model used least squares or an altered version of it for parameter estimation [10, 16–19]. Least squaresworked well because the noise affecting the data that is required by the momentum model is usually quite low. Despite6 able 1 A comparative analysis of model-based frameworks for whole-system parameter identification of aspace robot
Approach Comparision criterionA B C D EAyusawa et al. [14] ✓ ✓ ✓ ✗ ✗
Yoshida et al [16] ✓ ✗ ✓ ✓ ✗
Christidi et al. [18] ✓ ✗ ✓ ✓ ✓
Xu et al. [20] ✓ ✓ ✗ ✗ ✗
Proposed work ✓ ✓ ✓ ✓ ✓
Comparison criterion description:
A: Framework applies to generic tree-type robotic systemsB: Estimated parameters are sufficient for motion planning and controlC: Framework does not need components consuming fuelD: Model does not require acceleration or torque measurementsE: Systematic computation of exciting trajectories for parameter estimationthe presence of gravity-gradient torque, momentum model with least squares estimation performed well, evident fromthe momentum reconstructed by the estimated parameters in [16].
III. Momentum Model in Linear Form of Standard Parameters
In the usual momentum model formulation of a space robot, the inertial parameters are grouped among themselvesand also with the kinematic data and geometric parameters [20, 21]. Thus, it is hard to conclude the identifiabilityof inertial parameters using such a momentum formulation. In this section, we formulate the momentum model for ageneric tree-type space robot in the linear form of standard parameters. We consider an individual link in space, groupits inertial parameters, and separate the kinematic data into a Link Kinematic Matrix (LKM). LKM transforms the linkparameter vector to the link’s momentum. By assembling LKMs of all the links, the system kinematic matrix (SKM)is obtained which linearly transforms the standard parameters of the space robot to its momentum.A tree-type robot consisting of n + n joints is considered. Without loss of generality,the joints are considered to be revolute with 1-DoF, which is often the case with space robots. One of the links isconsidered as a base link, which is a satellite or spacecraft in space robots. Further, body frames are assigned to thelinks based on the modified Denavit-Hartenberg (DH) convention. The base link is equipped with a sensor to measureits linear velocity and euler rates, shown in Figure 1. To model the system, both linear and angular momentum areconsidered because angular momentum conservation does not necessarily imply linear momentum conservation andvice versa. 7 ig. 1 A dual-arm space robot with some of the notations. The sensor on the base link is at r in Σ I A. Linear Momentum of the i th Link
It is well-known that the linear momentum is linear in terms of the link parameters [17, 19] and for the i th link ( L i ),it is as follows: p i = m i Û c i (1)From the rigid body kinematics, the position and linear velocity of link CoM frame (cid:0) Σ ( c ) i (cid:1) and link frame ( Σ i ) arerelated as follows: c i = r i + a i (2) Û c i = v i + ωωω i × a i (3)Substituting Eq. (2) and Eq. (3) in Eq. (1) and separating the link parameters result in the link’s linear momentum as: p i = v i m i + ( ˜ ω ˜ ω ˜ ω i I R i ) m ii a i (4)where, ˜ ωωω represents the skew-symmetrix matrix representation of ωωω .˜ ωωω = − ω z ω y ω z − ω x − ω y ω x . Angular Momentum of the i th Link
For space robots, multiple works have used the nonlinear form of angular momentum for parameter estimation asin [16, 17, 20]. However, with insights from the linear form of the dynamic model [14, 15], we manipulate the angularmomentum model on the similar lines to arrive at the linear form. The angular momentum of L i is as follows: l i = I R ii I ( c ) i I R Ti ωωω i + c i × m i Û c i (5)Expanding Eq. (5) using Eq. (2) and Eq. (3) and denoting the cross product with skew-symmetric notation for thekinematic data, we get: l i = I R ii I ( c ) i I R Ti ωωω i + m i (cid:0) ˜ r i v i + ( ˜ r i ˜ ω ˜ ω ˜ ω i − ˜ v i ) a i + ( a i × ωωω i × a i ) (cid:1) (6)Note that in Eq. (5) and Eq. (6), the angular momentum is in nonlinear form of CoM parameters. However, it is linearin terms of mass and inertia. So, rewriting the nonlinear term a i × ωωω i × a i as a Ti a i E − a i a Ti to get: l i = (cid:16) I R ii I ( c ) i I R Ti + m i ( a Ti a i E − a i a Ti ) (cid:17) ωωω i + m i (cid:0) ˜ r i v i + ( ˜ r i ˜ ω ˜ ω ˜ ω i − ˜ v i ) a i (cid:1) (7)Substituting, a i = I R ii a i in Eq. (7). As a result of the parallel axis theorem, nonlinear CoM terms are eliminated bygrouping them with inertia around Σ ( c ) i resulting in inertia around Σ i as follows: l i = (cid:0) I R ii I i I R Ti (cid:1) ωωω i + m i (cid:0) ˜ r i v i + ( ˜ r i ˜ ω ˜ ω ˜ ω i − ˜ v i ) I R ii a i (cid:1) (8)where, i I i = i I ( c ) i + m i ( i a Ti i a i E − i a ii a Ti ) Eq. (8) is in the linear form of link parameters. Rewriting Eq. (8) by separating the link parameters from the kinematicdata as follows: l i = (cid:16) I R i (cid:2) ( I R Ti ωωω i ) • (cid:3) (cid:17) (cid:2) • i I i (cid:3) + (cid:0) ˜ r i v i (cid:1) m i + (cid:0) ( ˜ r i ˜ ω ˜ ω ˜ ω i − ˜ v i ) I R i (cid:1) m ii a i (9)where, [• I ] = (cid:20) I I I I I I (cid:21) T , [ ωωω •] = ω x ω z ω y ω y ω z ω x ω z ω y ω x Finally, Eq. (9) is in the desirable linear form of the link parameters.9 . System’s Total Momentum in Linear Form
Total momentum of L i is obtained by appending the linear and angular momentum equations, i.e, Eq. (4) and Eq.(9) into a matrix-vector product form as follows: K i φφφ i = p i l i (10)where, K i = O v i ˜ ω ˜ ω ˜ ω i I R iI R i (cid:2) ( I R Ti ωωω i ) • (cid:3) ˜ r i v i ( ˜ r i ˜ ω ˜ ω ˜ ω i − ˜ v i ) I R i ∈ R × φφφ i = (cid:20) (cid:2) • i I i (cid:3) T m i m ii a Ti (cid:21) T ∈ R × The matrix K i is the LKM, which consists of only the kinematic data of L i . The momentum of the entire system isobtained by assembling the LKMs of all the links resulting in a System Kinematic Matrix (SKM) denoted by K . K φφφ = pl (11)where, K = (cid:20) K . . . K i . . . K n (cid:21) ∈ R × ( n + ) φφφ = (cid:20) φφφ T . . . φφφ Ti . . . φφφ Tn (cid:21) T ∈ R ( n + )× The linear system of equations in Eq. (11) is evaluated for kinematic data (base pose and twist, joint angle andvelocity) at multiple instants and vertically appended together. The resultant is a global kinematic matrix (GKM)denoted by G . G φφφ = m (12)where, G = (cid:20) K ( t ) T . . . K ( t N m ) T (cid:21) T ∈ R N m × ( n + ) m = (cid:20) p ( t ) T l ( t ) T . . . p ( t N m ) T l ( t N m ) T (cid:21) T ∈ R N m × φφφ ). Even with sufficient data, all the standardparameters are individually not identifiable because some of the columns of GKM are linearly dependent. Under theaction of external force and torque on the space robot, the linear dependency in GKM is one of the following types: A)due to the kinematic constraints posed by the joints B) due to the choice of trajectory for constructing the GKM. Evenfor a fixed-base robot, linear dependencies occur due to same reasons with the matrix formulated using the linear formof dynamic model [26, 28]. Moreover, for a fixed-base robot, there are additional constraints due to its fixed base.Type-A linear dependency is inherent to the GKM and exists irrespective of the robot, as long as 1-DoF jointsconnect the rigid links. Type-A linear dependency is not avoidable and has to be eliminated. Type-B is because ofthe chosen joint trajectories and the response of the base link to them. Type-B linear dependency is avoidable, unlikeType-A. In the next two sections, we overcome both the types of linear dependencies in the momentum model. IV. Minimal Parameters and Regressor Matrix
A joint connects two adjacent links in a robot and constraints their kinematics, i.e., their physical motion. In thissection, we enforce these kinematic constraints onto the momentum model. Specifically, columns in GKM whichare linearly dependent due to the kinematic constraints are identified. From the set of linearly dependent columns,we obtain a set of basis columns which are linearly independent. Rest of the columns are called redundant columns,which are a linear combination of the basis columns with the coefficients of linear combination. Further, Type A lineardependency is eliminated by deleting the redundant columns from the GKM resulting in the regressor matrix. Usingthe coefficients of linear combination, the standard parameters are regrouped, resulting in the minimal parameters.Grouping the standard parameters compensates for the deleted redundant columns in the momentum model. Hence,the momentum model stays intact, and it is in the minimal form. More properties of minimal parameters are discussedin [12, 13].On a case-by-case basis, the redundant columns and the linear dependency coefficients can be computed usingmatrix factorization techniques on the numerical GKM for a particular robot similar to [23, 29]. We do not take suchan approach because the identified linearly dependent columns offers no distinction between the columns of Type-Aand Type-B. However, it is necessary to know the redundant columns due to Type-A exclusively, which should beeliminated. Exclusive knowledge of Type-A facilitates distinguishing Type-B linear dependency. Thus, the occurrenceof Type-B linear dependency can be avoided.We consider two adjacent links in a tree-type robot to find the redundant columns and their coefficients of linearcombination with the basis columns. Using these coefficients, a recursive grouping of parameters is performed toobtain the minimal parameters of a tree-type space robot. Two links connected by a 1-DoF joint have a parent-child11elationship. A parent is connected to the base link with a lower number of links than its child link. Each child hasonly one parent, and a parent can have multiple children. Base link has children, but it has no parent. Terminal linksdo not have children but have a parent.We use the following notation to represent the above mentioned relations between the links. For a child link withindex j , j ∈ I , j > , j ≤ n +
1, its parent is represented by ρ j ( ρ j < i ). Further, for a parent with index i , the set ofits children is represented by ζ i . The set of indices of all terminal links of a tree-type system is denoted by T such that ∀ e ∈ T , ζ e = {} .For the purpose of presentation, writing K i from (10) as follows: K i = (cid:20) K ( ) i K ( ) i K ( ) i (cid:21) (13)where, K ( ) i = (cid:20) O (cid:0) I R i (cid:2) ( I R Ti ωωω i ) • (cid:3) (cid:1) T (cid:21) T ∈ R × K ( ) i = (cid:20) v Ti ( ˜ r i v i ) T (cid:21) T ∈ R × K ( ) i = (cid:20) ( ˜ ω ˜ ω ˜ ω i I R i ) T (( ˜ r i ˜ ω ˜ ω ˜ ω i − ˜ v i ) I R i ) T (cid:21) T ∈ R × Consider a parent-child pair i , j such that ρ j = i . The relation between the pose and twist of links i , j due to arevolute joint is as follows: r j = r i + b j (14) I R j = I R ii R j (15) v j = v i + ωωω i × b j (16) ωωω j = ωωω i + I R j (cid:20) Û q j (cid:21) T (17)where b j = I R ii b j and Û q j is the joint velocity of the j th jointEq. (14) to Eq. (17) are substituted in LKM of child link, i.e., K j , and the linearly dependendent columns of K i and K j are identified using their symbolic forms. Consequently, the following redundant columns and their coefficients oflinear combination are obtained as follows: K ( ) j (cid:10) (cid:11) = K ( ) i t , j − K ( ) j (cid:10) (cid:11) (18) K ( ) j = − K ( ) i h • i ˜ b j i + K ( ) i + K ( ) i i b j (19)12 ( ) j (cid:10) (cid:11) = K ( ) i t , j + K ( ) i t , j (20)where, for a matrix A , A (cid:10) k (cid:11) represents k th column of At , j = (cid:20) , C α j , S α j , , C α j S α j , (cid:21) T t , j = (cid:20) i b ( ) j C α j − i b ( ) j S α j , i b ( ) j C α j , − i b ( ) j S α j , i b ( ) j S α j , i b ( ) j S α j − i b ( ) j C α j , − i b ( ) j C α j (cid:21) T t , j = (cid:20) , − S α j , C α j (cid:21) T C α = cos ( α ) , S α = sin ( α ) The choice of redundant columns is not unique [13]. However, we choose columns with the highest index, in linewith the traditional literature [30]. The redundant columns are 2 nd , 7 th , and 10 th columns of LKM of the child linkand correspond to the link parameters, i I ( yy ) i , m i , and m ii a ( z ) i . The linear dependency coefficients are obtained for anarbitrary parent-child link pair connected by a 1-DoF revolute joint. The redundant columns and the coefficients oflinear dependency obtained in Eq. (18) to Eq. (20) are in coherence with those obtained using the dynamic model in[30]. However, in [30] additional constraints due to the fixed base also exist. Also, it is clear that the obtained lineardependency coefficients are only a function of the geometric parameters. Hence, the linear dependency holds goodbetween LKMs of every parent-child pair in a tree-type system at every instant, irrespective of the joint trajectory.Further, it can be inferred from Eq. (18) to Eq. (20) that a parent-child link pair creates redundant columns in theLKM of the child. A base link has no parent. As a result, columns of LKM of the base link are linearly independentdue to kinematic constraints posed by joints. However, LKMs of the base link’s children have redundant columns.Since a terminal link does not have children, it does not contribute to the formation of redundant columns in LKM ofother links. However, terminal links have a parent in which they are always the child. Hence, the LKM of terminallinks have redundant columns. Rest of the links become both parent and child to different links. Hence, LKM of sucha link has redundant columns and creates redundant columns in the LKM of their child links.Based on the above-discussed logic, the links are divided into three categories, namely, link with no parent, linkswith no children, links with both parent and children. A recursive grouping of standard parameters corresponding toredundant columns is performed using the coefficients of linear dependency as follows:13or all links, e ∈ T (set of indices of terminal links)˜ φ ˜ φ ˜ φ e = φφφ e ˜ φ ˜ φ ˜ φ ( ) e = φφφ ( ) e − φφφ ( ) e φφφ m , e = D ( ˜ φ ˜ φ ˜ φ e ) K m , e = D ( K e ) (21)For all links, i < T , i ∈ I , i > , i ≤ n ˜ φ ˜ φ ˜ φ i = φφφ i + Õ ∀ j ∈ ζ i (cid:0) ˜ φ ˜ φ ˜ φ ( ) j k , j + ˜ φ ˜ φ ˜ φ ( ) j k , j + ˜ φ ˜ φ ˜ φ ( ) j k , j (cid:1) ˜ φ ˜ φ ˜ φ ( ) i = ˜ φ ˜ φ ˜ φ ( ) i − ˜ φ ˜ φ ˜ φ ( ) i φφφ m , i = D ( ˜ φ ˜ φ ˜ φ i ) K m , i = D ( K i ) (22)For the base link, φφφ m , = φφφ + Õ ∀ j ∈ ζ (cid:0) ˜ φ ˜ φ ˜ φ ( ) j k , j + ˜ φ ˜ φ ˜ φ ( ) j k , j + ˜ φ ˜ φ ˜ φ ( ) j k , j (cid:1) (23)where, φφφ m , ∈ R × , φφφ m , i ∈ R × ∀ i ∈ I + , k , j , k , j , k , j ∈ R × and k , j = (cid:20) t T , j (cid:21) T k , j = (cid:20) − (cid:2) • i ˜ b j (cid:3) T i b Tj (cid:21) T k , j = (cid:20) t T , j t T , j (cid:21) T The function D in Eq. (21) to Eq. (23) deletes the redundant columns from LKM, and the standard parameterscorresponding to the redundant columns from link parameter vector. Note that the recursion process to obtain theclosed-form solution of minimal parameters requires the knowledge of only the geometrical parameters.Upon performing the recursive grouping process, we obtain the minimal SKM ( K m ) and the minimal parametervector ( φφφ m ) which are: K m = (cid:20) K m , . . . K m , n (cid:21) ∈ R ×( + n ) (24) φφφ m = (cid:20) φφφ Tm , . . . φφφ Tm , n (cid:21) T ∈ R ( + n )× (25)14reviously, minimal parameters were obtained for a fixed-base robot using symbolic and numerical techniquesbased on the energy and dynamic models [13, 26, 30]. Further, the dynamics based formulation is also extendedto space robots [14]. Recently, a minimum set of parameters were obtained using only the angular dynamics andmomentum model [18]. However, to our best knowledge, this is the first time minimal parameters are systematicallyderived based on both linear and angular momentum model of space robots. This reformulation of momentum modelin terms of the minimal parameters forms the primary contribution of this work.The momentum model in Eq. (12) is rewritten using the regressor matrix ( G m ) and φφφ m as follows: G m φφφ m = m (26)where, G m = (cid:20) K m ( t ) T . . . K m ( t N m ) T (cid:21) T ∈ R N m ×( + n ) . The momentum ( m ) in Eq. (26) is applied by thereaction wheels. Note that we have leveraged the conservation of momentum principle by conserving the summationof momentum of space robot and the momentum applied by the reaction wheels onto the space robot. The system ofequations in Eq. (26) with sufficient data points (6 N m > + n ) is overdetermined. A least squares estimate of theminimal parameter vector ( ˆ φ ˆ φ ˆ φ m ) is: ˆ φ ˆ φ ˆ φ m = ( G Tm G m ) − G Tm m (27)Since the obtained minimal parameters fully construct the momentum model these parameters are sufficient formotion planning and momentum model-based control. Moreover, these minimal parameters are identical to the minimalparameters of floating-base robots based on the dynamic model in [14]. Hence, estimating these parameters not onlyallow for momentum-based control but also for dynamic model-based control. In conclusion, they are sufficient forkinematics and dynamics analysis, motion planning, and control.In this section, redundant columns due to Type-A linear dependency were identified and eliminated; thus, con-structed the minimal parameter vector and regressor matrix. Next, we avoid the occurrence of Type-B linearlydependency in the regressor matrix. V. Exciting Trajectories
We present the other contribution of this work, which is the joint trajectory planning and optimization frameworkfor computing exciting trajectories. First, an overview of the entire framework is provided, following which the methodsand modules within the framework are discussed.
A. Outline of the Proposed Framework
The exciting trajectory computation framework majorly consists of joint actuation strategy, trajectory parametriza-tion, optimization, and pruning modules, as shown in Figure 2. The geometric parameters of the robot and an initial15 ig. 2 Block diagram of joint trajectory planning and exciting trajectories computation guess of the inertial parameters are the inputs to the framework, and they can be obtained from the robot’s CAD model.Based on the geometric parameters of the robot, the joint actuation strategy decides the direction of joints’ velocityduring a time interval, which is encapsulated in direction combinations. Each interval executes a unique directioncombination and consequently excites a subset of minimal parameters. The entire trajectory is composed of multipleintervals and thus excites all the minimal parameters, rendering them identifiable. The joint trajectory within a specificinterval is parameterized with the interval trajectory parameters. In turn, the interval trajectory parameters of all theintervals of a particular joint are parameterized by the seed parameters. The seed parameters along with the knowledgeof direction combinations parameterize the entire trajectory, hence, they are the design parameters of the trajectoryoptimization problem. Specifically, computing the exciting trajectories requires the following three steps: • Step 1: Parameterization of the joint trajectories with all the direction combinations and trajectory optimization,resulting in intermediate exciting trajectories • Step 2: Pruning the intervals of the intermediate exciting trajectories, resulting in optimal direction combinations • Step 3: Parameterization of the joint trajectories with only the optimal direction combinations, and trajectoryoptimization, resulting in optimal seed parameters and the exciting trajectoriesIn the block diagram in Figure 2, the inner loop containing the
Joint trajectories and
Seed parameters is executedin both Step 1 and Step 3 until the objective function converges to minima. The outer loop containing the
Trajectorypruning is executed only once, i.e., in Step 2. Next, we discuss the design of each module.
B. Joints’ Actuation Strategy
We propose to excite the minimal parameters of a space robot using direction combinations of joint rates. Adirection combination is a set of directions of the joint velocity of all the joints at a given instant. A non-zero jointvelocity has two directions, either positive or negative. We fix the direction of the joint velocity of one joint constant16hroughout the entire duration. As a result, for a system with n joints, there are 2 n − direction combinations. Eachdirection combination is executed in a specific time interval.In [31], due to the inherent nature of B-splines, the exciting trajectory is composed of multiple intervals. However,in this work, the exciting trajectory is purposely composed with multiple intervals because each interval, composedwith a unique direction combination, excites a subset of minimal parameters. C. Interval Trajectory Parametrization
Here, parametrization of joint position and joint velocity in a time interval is discussed. The proposed intervaltrajectory parametrization facilitates achieving the joint actuation strategy.A cycloidal function in time parameterizes the joint position. It requires just two parameters ( θ kj f , θ kji ) to producecontinuous and differentiable joint position, joint velocity, and joint acceleration. For j th joint during k th interval, j , k ∈ I , j , k > j ≤ n , and k ≤ n − , the joint position and joint velocity are: q kj = θ kji + ( θ kj f − θ kji ) T p " t − T p π sin π tT p ! (28) Û q kj = ( θ kj f − θ kji ) T p " − cos π tT p ! (29)Along with having only two parameters to parameterize the trajectory in an interval, cycloidal parametrization alsooffers following advantages: • Position function is monotonic in a given interval. The two parameters of the cycloidal function physically arethe initial and final positions of the joint in an interval. Hence, by simply bounding those two parameters, thejoint position can be contained within the desired limits. • Since the position function is monotonic in a given interval, the joint velocity has a single direction in an interval.This behavior is extremely advantageous to execute the velocity combinations. By choosing a final positiongreater than initial position, a positive joint velocity is obtained. Similarly, for negative joint velocity, the finalposition should be lesser than the initial position. • Further, analytical solutions exist for maxima and minima on velocity and acceleration and they result in linearconstraints. Thus, constraining them within limits is simplified.The joint trajectory is obtained by concatenating the interval trajectories. Further, the final position of the previousinterval is the initial position of the current interval. Hence, for a single joint, f intervals result in f + n joints, there are 2 n − intervals. Consequently, for n joints, there are a totalof n ( n − + ) interval trajectory parameters. 17 . Seed Parameters The interval trajectory parameters parameterize an interval trajectory. Here, seed parameters are introduced, whichparameterize the interval trajectory parameters of all the intervals of a joint. The total number of interval trajectoryparameters grows as n ( n − + ) , which is undesirable for optimization and does not scale with increasing dimensionsof the robot. Even with just six joints, the design space of optimization is of 198 dimensions. Hence, a small numberof seed parameters are proposed to parameterize the ( n − + ) interval trajectory parameters of a joint.For each joint, a seed consists of just two parameters, namely, start and range parameters. Start parameter is theinitial position of the first interval. The range parameter is either added or subtracted to the initial position of an intervalto get the final position. Addition or subtraction of range is decided by the direction requirement of the joint velocityin that interval. If a positive velocity is required, the range term is simply added, else subtracted. For j th joint during k th interval, the seed parameters construct the trajectory parameters as follows: θ kji = θ js , k = θ k − j f , k > θ kj f = θ k − j f + θ jd , Û q kj > θ k − j f − θ jd , Û q kj < θ js and θ jd are the start and range parameters of the j th joint respectively. Since θ jd is range, it is alwaysgreater than zero. A zero range leads to a constant joint position resulting in no joint motion. The interval trajectoryparameters constructed in Eq. (30) and Eq. (31) are used to compute interval trajectories using Eq. (28) and Eq. (29).Using a seed, the design parameters of the optimization problem drastically reduce from n ( n − + ) to 2 n . Sections V.C and V. D together comprise the Trajectory Parametrization block in Figure 2.
E. Trajectory Optimization
The trajectory optimization problem is discussed in this subsection, which results in the optimal values of theseed parameters. The primary objective of exciting trajectories is to render all the minimal parameters identifiable.Specifically, the trajectory should not cause linear dependency (Type-B) in the columns of regressor matrix. Further,the sensor measurements are corrupted by noise. Hence, the secondary objective is to ensure that the estimates of theminimal parameters are minimally affected by noise. Exciting trajectories are computed by minimizing a cost functionthat fulfills the desired objectives. 18 . Objective Function
We define an objective function involving the condition number of the regressor matrix and the magnitude of thebase velocities and joint rates. min θ js ,θ jd ∀ j ∈ n C = κ ( G m ) + w N t N m Í N t i = Í N m j = h ij (32) h is the joint velocity or the component of base twist.A low condition number ensures that the trajectory uniformly excites all the minimal parameters and results in theestimates which are relatively insensitive to noise. Moreover, the inverse of squares of base velocities and joints rates areadded to the objective function to achieve trajectories measurements with high SN ratio. Since the measurement noise isindependent of the measured value, higher velocity measurements correspond to a higher SN ratio. Measurements withlow SN ratio do not represent the true behavior of the system, and the accuracy of estimates could be adversely affected.Both components of the objective function together help in realizing both the objectives of exciting trajectories.
2. Simplified Joint Constraints
The proposed interval trajectory parametrization with cycloidal trajectories and further parametrization of theinterval trajectory parameters with the seed parameters not only reduces the number of design parameters of theoptimization problem but also simplifies the joint constraints. We derive a simplified set of constraints, which result intrajectories adhering to joint angle, joint velocity, and joint acceleration limits.The joint position can be simply constrained by bounding the interval trajectory parameters within the limits asfollows: θ min j ≤ θ kji , θ kj f ≤ θ max j (33)Given the direction combinations, the interval trajectory parameters are computed from the seed parameters with onlylinear operations. Hence, the constraint in Eq. 33 leads to only linear constraints on the design parameters of theoptimization problem.The maximum of the magnitude of j th joint velocity in k th interval is max (|| Û q kj ||) = ( θ kj f − θ kji )/ T p . However, forall k , ||( θ kj f − θ kji )|| = θ jd , see Eq. (30) and Eq. (31). Hence, max (|| Û q kj ||) = θ jd / T p . To adhere to the joint velocitylimit, 2 θ jd / T p ≤ min (|| Û θ min j || , || Û θ max j ||) . On rearranging the equation, we get θ jd ≤ ( T p / ) min (|| Û θ min j || , || Û θ max j ||) (34)Similarly, maximum of the magnitude of j th joint acceleration in k th interval can also be obtained, which ismax (|| Ü q kj ||) = πθ jd / T p . To adhere to the joint acceleration limit, 2 πθ jd / T p ≤ min (|| Ü θ minj || , || Ü θ maxj ||) . On rearranging19he equation, we get θ jd ≤ ( T p / π ) min (|| Ü θ min j || , || Ü θ max j ||) (35)To satisfy both the Eq. (34) and Eq. (35), θ jd ≤ min (cid:16) ( T p / ) min (cid:0) || Û θ min j || , || Û θ max j || (cid:1) , ( T p / π ) min (cid:0) || Ü θ min j || , || Ü θ max j || (cid:1) (cid:17) (36)The right hand side of Eq. (36) is simply a constant, represented by θ max jd , which can be computed from the jointlimits. Since, θ jd >
0, the bound constraint on θ jd , ∀ j ∈ I , j > , j ≤ n is as follows:0 < θ jd ≤ θ max jd (37)The proposed parametrization leads to only linear and bound constraints on the design parameters of the optimizationproblem for satisfying the joint limits. The 2 n design parameters are obtained by minimizing the objective function inEq. (32) with Eq. (33) and Eq. (37) as the constraints. Sequential Quadratic Programming (SQP) is used to solve theconstrained optimization problem, which allows to include nonlinear constraints also. F. Pruning the Intermediate Exciting Trajectories
Even though 2 n − intervals add unique data about the system, not all of them are required to render the parametersidentifiable and well-condition the regressor matrix. We present a methodology to prune such intervals. Each intervalhas a unique direction combination; pruning the intervals of intermediate exciting trajectories results in the optimaldirection combinations. Moreover, with all the 2 n − intervals, scaling the methodology to high-dimensional systemsbecomes infeasible because the time required for them to execute the exciting trajectories grows exponentially.To perform pruning (Step 2), we first compute the intermediate exciting trajectories, which consider all the directioncombinations (Step 1). The process of pruning proceeds as shown in Algorithm 1. Regressor matrix ( G m ) is constructedusing intermediate exciting trajectories, which is an input to pruning algorithm. Next, the regressor matrix constructedwith the trajectory data of unpruned intervals is represented by ( G om ). Given the regressor matrix ( G m ) and the intervalindex ( i ), getIntervalRM retrives all the rows of G m corresponding to the i th interval index represented by G im . Firstfew intervals are unpruned until G om has higher number of rows than columns ( n r > n c ).Next, every G im is vertically concatenated to the G om , resulting in G tm . The properties of G tm are compared with G om to check if appending the regressor matrix of a particular interval has helped in achieving the objectives of computingexciting trajectories. Finally, i th interval from the rest of the intervals is unpruned if one of the following conditions issatisfied by G tm :1) The number of significant singular values of G tm , computed by the function numSignificantSingularVals , is20 lgorithm 1: Prune intervals
Input : G m , n , u max , δ Output : g i , n r , n c , u ← g , G om ← [ ] while n r ≤ n c do i ← i + g ← [ g , i ] G im ← getIntervalRM ( G m , i ) G om ← [( G om ) T , ( G im ) T ] T [ n r , n c ] ← size ( G om ) end G tm ← G om s prev ← numSignificantSingularVals ( G om , δ ) cn prev ← condNum ( G om ) while i < n − do i ← i + G im ← getIntervalRM ( G m , i ) G tm ← [( G om ) T , ( G im ) T ] T s now ← numSignificantSingularVals ( G tm , δ ) cn now ← condNum ( G tm ) if (cid:16) ( s now > s prev ) or ( cn now < cn prev ) or (cid:0) ( s now == s prev ) and ( u ≤ u max ) (cid:1)(cid:17) then g ← [ g , i ] G om ← G tm if (cid:0) ( s now == s prev ) and ( u ≤ u max ) (cid:1) then u ← u + else u ← end else u ← end end more than that of G om , i.e., s now > s prev .2) The condition number of G tm , computed by the function condNum , is less than that of G om , i.e., cn now < cn prev .3) The number of significant singular values of G tm are equal to G om and occurs continuously for not more than u max times.Significant singular values are determined by the number of singular values whose magnitude is greater thana fraction ( δ ) of the singular value with the highest magnitude. First condition means that unpruning a particularinterval excites higher number of minimal parameters than without it. Second condition means that the system iswell-conditioned with that particular interval than without it. Third condition is applied to reduce the effect of initialguess on the exciting trajectories. The index of every unpruned interval is stored in g whose elements are the indicesof optimal direction combinations. Using the optimal direction combinations, the trajectory is parameterized and seed21arameters are optimized, i.e. Step 3, also shown in Figure 2. The trajectory parameterized with the optimal directioncombinations and the optimal seed parameters are the exciting trajectories to be executed on the system of interest. VI. Results and Discussion
In this section, we numerically apply the proposed identification framework on a spatial space robot. The excitingtrajectories are computed, and they are executed by the space robot. The kinematic data of the robot is capturedwhile performing the exciting trajectories, using which all the minimal parameters are estimated. Furthermore, theperformance of the estimated minimal parameters is evaluated by performing inverse dynamics computations tocalculate the error in the predicted base twist and joint torques in comparison to the ground truth.
A. Space Robot Setup
With a growing interest in tasks like active debris removal and refurbishment of retired satellites, multi-arm spacerobots are expected to grow. Hence, we demonstrate the efficacy of the proposed framework by identifying the minimalparameters of a 12-DoF, spatial, dual-arm space robot, shown in Figure 3. Each arm has three links connected by1-DoF revolute joints. Two such arms contribute to 6-DoF. The base itself has 6-DoF resulting in a 12-DoF spacerobot. The true inertial parameters of the space robot are given in Table 2. Terminal links, namely, L , L , hold ageneric payload. Hence, their true inertial parameters are distributed along all the axes and higher than other links ofthe robotic arms. Further, the robot is equipped with three reaction wheels mounted on the base link along mutuallyorthogonal axes. However, the proposed identification framework applies to other arrangements of reaction wheelsalso. The closed-form solution of minimal parameters obtained using Eq. (21) to Eq. (23) for the robot shown inFigure 3 are available in the Appendix A. 1.Base Link Joint 1 Joint 2Joint 3Joint 4Joint 5Joint 6 x y z Fig. 3 The architecture of the 12-DoF, spatial, dual-arm space robot considered for the numerical study. Thereaction wheels are mutually orthogonal with 3-DoF. able 2 True inertial parameters of the robot and the offset inertial parameters. The offset inertial parametersare used to generate the exciting trajectories for minimal parameter identification Parameters L L L L L L L Mass(Kg)
T 2000 50 40 30 50 35 60O 1500 30 20 50 40 25 30 x-CoM(m)
T 0.2 0.6 0.4 0.7 0.55 0.45 0.6O 0 0.4 0.7 0.5 0.4 0.3 0.5 y-CoM(m)
T 0.3 0.05 -0.45 0.4 0.04 0.05 -0.5O 0 0 0 0.2 0 0 0.2 z-CoM(m)
T 0.4 -0.07 -0.05 0.3 -0.04 0.05 -0.35O 0 0 0 0.1 0 0 0.2 I cx x (Kg-m ) T 1200 3.1 1.15 24.45 1.85 2.55 12.24O 1000 2 1 14 1 1 6 I cyy (Kg-m ) T 1200 1.89 1.68 28.56 1.62 1.84 31.45O 1000 1 1 20 2 1.5 20 I czz (Kg-m ) T 1200 20.51 18.67 35.53 17.05 14.28 23.77O 1000 10 28 20 9 19 35 I cxy (Kg-m ) T 35.52 1.9 0.61 9.78 1.5 2.9 9.1O 25 0.9 1.2 4.6 1 1.5 15 I cyz (Kg-m ) T 40.45 3.65 1.75 9.1 2.25 1.55 8.52O 20.45 2.65 2.75 3.1 1.25 0.55 18.52 I cz x (Kg-m ) T 45.71 3.9 1.5 10.23 3.71 1.27 8.67O 65.71 2.9 2.5 1.23 6.71 4.27 4.67T and O represent true and offset inertial parameters respectively
B. Exciting Trajectories
Here, we discuss the details required to compute exciting trajectories, described in Figure 2, while providinginsights into the proposed exciting trajectories computation framework.
1. Direction Combinations
There are six joints in the space robot and three in the reactions wheels resulting in a total of nine joints. Hence,the number of direction combinations are 2 − =
2. Reaction Wheels Momentum
Each reaction wheel has predetermined seed parameters. The start and range parameters as 0 rad and 3 π radrespectively. They produce a maximum angular momentum of 18.8496 Kg-m /sec along its axis of rotation in its body23rame. In case of a different reaction wheels arrangement, the entire framework remains the same. However, the otherreaction wheels arrangement should execute the momentum trajectories computed using three mutually orthogonalwheels arrangement.
3. Intermediate Exciting Trajectories
The objective function in Eq. (32) is minimized using all the 256 direction combinations to obtain the intermediateexciting trajectories. Minimizing the cost requires kinematic data of the base link and the joints. Since base motionis a response to the joint trajectories according to the robot’s momentum model, computing the cost requires an initialguess of the minimal parameters.For physical systems, an initial estimate is available from the CAD models of the system. However, to check therobustness of the methodology, we offset the true inertial parameters within ±
20% - 120% of their values, shown asoffset parameters in Table 2, and pass them as an initial guess to the exciting trajectory computation framework. Notethat the guess of the terminal links and the base link are highly offset because on-orbit operations substantially modifythe CAD model’s estimates.
4. Trajectory Pruning and Exciting Trajectories
Out of 256 intervals in the intermediate exciting trajectories, 189 of them are pruned using Algorithm 1. u max and δ in Algorithm 1 are chosen as 5 and 1/300 respectively. Hence, the number of singular values whose magnitude isgreater than 1/300 times the magnitude of the highest singular value is the number of significant singular values. Asa result, 67 unique optimal direction combinations are obtained, which constitute the exciting trajectories. The resultof pruning suggests that the proposed exciting trajectories computation framework is scalable with increasing DoFbecause only some of the 2 n − direction combinations are required. The joint trajectories are parameterized with the 67optimal direction combinations, and the seed parameters are optimized resulting in exciting trajectories. The optimalseed parameters of the exciting trajectories are tabulated in Table 3.
5. Analysis of the Exciting Trajectories Computation Framework
For analysis, using the exciting trajectories, the condition number of the regressor matrix and the number ofsignificant singular values are computed for the true system, shown in Figure 4. These computations are made for twokinds of trajectories, a) interval trajectory, which consists of one particular direction combination and b) concatenatedtrajectory, which consists of several interval trajectories appended together up to the interval index.Each interval trajectory excites only some minimal parameters of the robot up to the desired level (in this case, 8to 11 out of 52). However, different interval trajectories excite a different subset of minimal parameters. Consequently,all the optimal interval trajectories together excite all the minimal parameters, apparent from Figure 4. Further, the24 nterval Index C ond iti on N u m b e r Interval Index N u m b e r o f s i gn i f i ca n t s i ngu l a r v a l u e s Concatenated trajectory Interval trajectory
Fig. 4 a) Condition number and b) Number of significant singular values of the regressor matrix obtainedusing exciting trajectories executed on the true system condition number of the regressor matrix constructed with a single interval is extremely high ( ∼ ) because it excitesonly some minimal parameters. However, optimal intervals and the optimal seed parameters have brought a significantreduction in the condition number to 248.8. The proposed concept of direction combinations resulted in intervaltrajectories. The interval trajectories together excited all the minimal parameters. Thus, the minimal parameters arerendered identifiable while also resulting in a low condition number of the regressor matrix.Further, the number of seed parameters are only 2 n with the proposed technique, 5.5 times lower than the trajectoryparameters of commonly used 5-term fourier series. Such a reduction in the number of design parameter for optimizationmakes it highly desirable to generate exciting trajectories even for high-dimensional space robots. Moreover, evenafter generating exciting trajectories with hugely offset inertial parameters, the trajectories excited all the minimalparameters and provided a regressor matrix with a low condition number. This indicates that the proposed frameworkfor computing exciting trajectories is robust to a bad initial guess of inertial parameters.In Figure 4, the condition number takes a massive drop twice. The first drop occurs between the 1 st and 3 rd intervals.The number of significant singular values because of first interval is 11. Upon appeding the 2 nd and 3 rd intervals, thenumber of significant singular values improve to 20. Owing to a sudden improvement in the number of significantsingular values led to a massive drop in the condition number. Such an improvement is possible initially because only11 of the 52 minimal parameters are excited by the 1st interval. The chances of exciting new minimal parameters by thesubsequent interval is high. The second drop takes place when number of significant singular values reach 52, whichis equal to the number of minimal parameters. Until then, at least one minimal parameter was not sufficiently excitedby the trajectory resulting in an ill-conditioned regressor matrix.25 able 3 Seed parameters of exciting joint trajectories obtained with offset inertial parameters as initial guessto the trajectory optimizer Seedparameters(deg)
Joint index1 2 3 4 5 6
Start ( θ s ) Range ( θ d ) C. Parameter Estimation
In this subsection, the exciting trajectories are executed by the space robot in ReDySim [32], and the kinematicdata is captured. Using the captured kinematic data, the regressor matrix is constructed and the minimal parametersare estimated using Eq. (27). It may be noted that, unlike the existing whole-system momentum-based identificationframeworks, we do not estimate only a few parameters of all the links. We estimate even the parameters including theCoM terms along the y-axis and z-axis and product of inertia of the links, which are ignored by the existing works.To check the validity of the proposed identification framework, the regressor matrix is constructed using thekinematic data that is free of noise, and the minimal parameters are estimated. Without noise, as expected, theestimates match exactly with the true values, tabulated in Table 4.For the simulated data to mimic the real data, zero-mean gaussian white noise is added to the joint rates and basevelocities, as in [17]. The standard deviation in measurement noise on linear and angular velocity components of thebase and joint rates are 50 µ m/sec, 80 µ rad/sec, and 50 µ rad/sec respectively. The regressor matrix is constructed withthe noisy kinematic data and the minimal parameters are estimated, shown under Case 1 in Table 4. To understandthe accuracy of the estimates, the median and maximum of the relative error of the estimated parameter estimates wrt.their true values are computed, and they are tabulated in Table 5. ǫ median = median i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φφφ ( i ) m − ˆ φφφ ( i ) m φφφ ( i ) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (38) ǫ max = max i = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φφφ ( i ) m − ˆ φφφ ( i ) m φφφ ( i ) m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (39)The median relative error is as low as 2%, and the maximum relative error is 174%. Most of the estimatedparameters are quite accurate apart from 10-11 parameters which have a relative error more than 10%. Such a relativeerror occurred with parameters having comparatively low numeric values because they have a low contribution towardsthe system behavior; hence, noise easily distorts their estimates. These are the minimal parameters containing the26roduct of inertia terms of the links of the robotic arm. However, terms containing the product of inertia of either thebase link or the end effector link are accurately estimated because their numeric value is comparatively high. Evenwith the obtained relative error, in the next subsection, we show that these parameters predict the future states of thespace robot accurately. Table 4 True and estimated value of minimal parameters of 12-DoF space robot with different joint trajectories
MinimalParameter TrueValue Estimated ValueCase 1 Case 2 Case 3 ˆ φφφ ( ) m [Kg-m ] 2130.560 2126.814 1729.274 2121.443ˆ φφφ ( ) m [Kg-m ] 2078.560 2071.702 1930.767 2109.644ˆ φφφ ( ) m [Kg-m ] 1555.700 1554.130 1801.419 1740.076ˆ φφφ ( ) m [Kg-m ] -96.630 -95.304 -875.382 -83.209ˆ φφφ ( ) m [Kg-m ] -269.450 -267.023 -607.177 -294.371ˆ φφφ ( ) m [Kg-m ] -148.890 -142.312 26.411 -103.739ˆ φφφ ( ) m [Kg] 2265.000 2244.299 1418.330 2198.283ˆ φφφ ( ) m [Kg-m] 440.500 436.558 800.561 440.037ˆ φφφ ( ) m [Kg-m] 679.500 673.810 574.592 658.626ˆ φφφ ( ) m [Kg-m] 1033.000 1023.873 621.876 1005.331ˆ φφφ ( ) m [Kg-m ] -2.525 -6.919 1305.865 -34.097ˆ φφφ ( ) m [Kg-m ] 192.775 188.387 909.389 -14.112ˆ φφφ ( ) m [Kg-m ] 7.400 10.166 614.689 4.942ˆ φφφ ( ) m [Kg-m ] 3.825 7.047 537.259 85.749ˆ φφφ ( ) m [Kg-m ] 6.000 2.818 -521.054 0.024ˆ φφφ ( ) m [Kg-m] 100.000 98.937 -555.623 90.001ˆ φφφ ( ) m [Kg-m] -4.500 -4.993 253.047 5.614ˆ φφφ ( ) m [Kg-m ] -36.866 -30.258 -112.659 -30.894ˆ φφφ ( ) m [Kg-m ] 55.134 56.976 240.184 62.47527 inimalParameter TrueValue Estimated ValueCase 1 Case 2 Case 3 ˆ φφφ ( ) m [Kg-m ] 1.250 1.915 2.599 3.191ˆ φφφ ( ) m [Kg-m ] 1.670 1.422 129.586 6.837ˆ φφφ ( ) m [Kg-m ] -6.700 -6.544 103.711 -12.255ˆ φφφ ( ) m [Kg-m] 46.000 45.310 74.935 45.001ˆ φφφ ( ) m [Kg-m] -1.600 -1.438 -47.935 -4.119ˆ φφφ ( ) m [Kg-m ] -14.010 -11.737 -28.911 -12.295ˆ φφφ ( ) m [Kg-m ] 55.030 54.600 69.948 52.819ˆ φφφ ( ) m [Kg-m ] 1.380 -0.975 -15.320 -1.341ˆ φφφ ( ) m [Kg-m ] 5.500 5.833 -4.728 5.395ˆ φφφ ( ) m [Kg-m ] 3.930 3.865 2.534 2.835ˆ φφφ ( ) m [Kg-m] 21.000 21.080 13.574 20.316ˆ φφφ ( ) m [Kg-m] 12.000 12.092 7.293 12.556ˆ φφφ ( ) m [Kg-m ] 19.600 19.097 84.010 12.528ˆ φφφ ( ) m [Kg-m ] 256.670 258.902 463.905 302.331ˆ φφφ ( ) m [Kg-m ] -18.850 -18.251 40.214 -19.746ˆ φφφ ( ) m [Kg-m ] 2.330 2.474 -477.795 11.882ˆ φφφ ( ) m [Kg-m ] 4.810 3.999 65.519 -8.016ˆ φφφ ( ) m [Kg-m] 122.500 121.080 -12.235 115.634ˆ φφφ ( ) m [Kg-m] 21.250 20.857 -179.302 19.762ˆ φφφ ( ) m [Kg-m ] -66.290 -67.589 -103.468 -74.876ˆ φφφ ( ) m [Kg-m ] 81.455 82.689 -387.648 81.355ˆ φφφ ( ) m [Kg-m ] 2.112 1.306 -12.430 2.533ˆ φφφ ( ) m [Kg-m ] 1.462 2.565 44.779 3.472ˆ φφφ ( ) m [Kg-m ] 21.483 20.749 206.239 20.47428 inimalParameter TrueValue Estimated ValueCase 1 Case 2 Case 3 ˆ φφφ ( ) m [Kg-m] 75.750 75.350 -25.984 74.843ˆ φφφ ( ) m [Kg-m] 1.750 1.832 54.533 2.982ˆ φφφ ( ) m [Kg-m ] -25.810 -25.265 -55.123 -21.904ˆ φφφ ( ) m [Kg-m ] 60.370 61.114 84.505 61.583ˆ φφφ ( ) m [Kg-m ] 27.100 27.657 38.051 27.333ˆ φφφ ( ) m [Kg-m ] -1.980 -2.074 -5.270 -1.736ˆ φφφ ( ) m [Kg-m ] 21.270 21.145 24.745 21.551ˆ φφφ ( ) m [Kg-m] 36.000 35.662 21.960 35.448ˆ φφφ ( ) m [Kg-m] -30.000 -29.627 -17.515 -28.990Further, we study the effect of the number of significant singular values of the regressor matrix on minimalparameter estimates. For this purpose, two trajectories with only the first 20 and 40 intervals are considered to estimatethe parameters. The minimal parameter estimates are tabulated for 20 and 40 intervals under Case 2 and Case 3respectively in Table 4. Median and maximum relative error in the parameters is calculated and tabulated in Table 5.The trajectory in Case 2 and Case 3 excites 38 and 46 out of the 52 minimal parameters respectively. In comparisonto Case 1, the median and maximum relative errors are substantially higher for both the Case 2 and Case 3 becauseall the minimal parameters are not excited sufficiently. Although 46 out of 52 minimal parameters are excited inthe trajectory with 40 intervals, the relative error is much higher compared to Case 1 reinforcing the idea that everyparameter has to be sufficiently excited by the trajectory. Such an excitation is thoroughly obtained by the proposedframework at a much lower computational effort even with a bad initial guess. Table 5 Relative error in the estimated minimal parameters wrt. their true values
Case 1 Case 2 Case 3 ǫ meadian ǫ max . Performance Evaluation of the Proposed Framework The primary goal of estimating the minimal parameters is to facilitate modeling, motion planning, and control ofa space robot. The motion planning and control algorithms rely on the kinematic and dynamic model of the system.So, we use the estimated parameters to evaluate the accuracy with which they predict the kinematics and dynamics ofthe system, as in [20, 33]. We evaluate the performance of the proposed framework by conducting inverse dynamicssimulations with the true and estimated minimal parameters and computing the error in base twist and joint torques asdepicted in Figure 5. To consolidate the error trajectory into a single parameter, its RMS is computed.
Fig. 5 Block diagram of the performance evaluation scheme. The joint trajectory inputs to the inverse dynamicsblocks are in Figure 6, and the prediction error outputs are in Figure 7.
Cycloidal parametrization is used for parameter estimation. Hence, for performance evaluation of the estimatedparameters, we consider a fourier series parametrization for joint position as follows: q j = q j , + N h Õ i = q ( s ) j , i sin ( i ω f t ) + q ( c ) j , i cos ( i ω f t ) (40) Û q j = N h Õ i = i ω f (cid:0) q ( s ) j , i cos ( i ω f t ) − q ( c ) j , i sin ( i ω f t ) (cid:1) (41)Here, N h = ω f = π /
20. The fourier based trajectories for joint angles and their rates are shown in Figure 6.The RMS of error for an arbitrary trajectory ( ψ ) is defined as follows: ǫ ( ψ ) = vut N m N m Õ i = ( ψ i − ˆ ψ i ) (42)We apply Eq. (42) to base twist and joint torque whose results are in Table 6. The error trajectories are shown in Figure7. From Table 6, it is clear that the RMS error in the predicted linear velocity is only 1 - 3 times higher than thestandard deviation of its measurement noise (5e-5 m/s) and that of angular velocity is only 1 - 4 times higher than thestandard deviation of its measurement noise (8e-5 rad/s). Further, the RMS error in predicted torque is much lower, i.e.,30 ig. 6 Fourier series based parametrization of joint angle and velocity trajectories used for performanceevaluation of the estimated parametersFig. 7 Error in the predicted base twist components and joint torques by 1 - 10 times the standard deviation of the usual measurement noise of joint torque (0.3808 Nm [33]). With such lowerror in predicting the kinematics and dynamics of the system, it is safe to conclude that the parameters estimated usingthe proposed identification framework facilitates accurate analysis, motion planning, and control of a space robot. VII. Conclusion
The applicability of the momentum model of tree-type space robots for minimal parameter identification is examinedin this work. Momentum model is linearly formulated in terms of the minimal parameters of a space robot. It is achievedby reformulating the generic momentum model and imposing the kinematic constraints posed by the space robot’sjoints. The minimal parameters uniquely model the space robot’s momentum and dynamic models; hence, they are31 able 6 RMS error in predicted base velocities and joint torques
Base twistcomponent v ( x ) v ( y ) v ( z ) ωωω ( x ) ωωω ( y ) ωωω ( z ) m/s m/s m/s rad/s rad/s rad/sRMS error 1.7e-4 1.6e-4 0.6e-4 2.1e-4 1.8e-4 3.5e-4 Jointtorque τ τ τ τ τ τ Nm Nm Nm Nm Nm NmRMS error 0.2961 0.1665 0.0778 0.0922 0.0368 0.0262sufficient for kinematic and dynamic analysis, motion planning, and control.A systematic framework for computing the exciting trajectories is proposed based on an interval-wise approach.Each interval is constructed using a unique direction combination of the joints’ velocity. A single interval excites asubset of minimal parameters; all the intervals excite the entire set of minimal parameters. The proposed frameworkis computationally efficient with only 2 n design parameters for optimization and also results in only linear and boundconstraints on the design parameters to adhere to the joint limits. Hence, the proposed framework is scalable withincreasing degrees-of-freedom. Even though an initial guess of inertial parameters is required for computing excitingtrajectories, it is shown that even a bad guess computed exciting trajctories that resulted in accurate parameter estimates.Further, the minimal parameters of a 12 degrees-of-freedom, spatial, dual-arm space robot are identified with thekinematic data corrupted by noise. The estimated parameters predicted the base motions and joint torques accuratelywith prediction errors mostly in the order of noise in their measurements. Such accurate predictions with the proposedmethodology even in the presence of noise facilitates accurate motion planning and control. The presented approachis generic and applicable to floating-base, tree-type robotic systems with reaction wheels. Consequently, it is alsoapplicable to the underwater robots under the conservation of momentum. Appendix
A. Analytical Expression of Minimal Parameters
The components of minimal parameter vector for the space robot shown in Figure 3 are as follows: φφφ ( ) m = I ( xx ) + I ( yy ) + I ( yy ) + . m + . m + . m + . m + . m + . m + . ( m a ( z ) + m a ( z ) ) φφφ ( ) m = I ( yy ) + I ( yy ) + I ( yy ) + . m + . m + . m + . m + . m + . m + . ( m a ( z ) + m a ( z ) ) φφ ( ) m = I ( zz ) + . ( m + m + m ) + . ( m + m + m ) φφφ ( ) m = I ( xy ) − . ( m + m + m ) + . ( m + m + m ) φφφ ( ) m = I ( yz ) − . ( M − m ) − . ( m a ( z ) + m a ( z ) ) φφφ ( ) m = I ( zx ) − . ( m + m + m ) + . ( m + m + m )− . m a ( z ) + . m a ( z ) φφφ ( ) m = M φφφ ( ) m = m a ( x ) + . ( m + m + m ) − . ( m + m + m ) φφφ ( ) m = m a ( y ) + . ( M − m ) φφφ ( ) m = m a ( z ) + m a ( z ) + m a ( z ) + . ( M − m ) φφφ ( ) m = I ( xx ) − I ( yy ) + I ( yy ) + I ( yy ) − m φφφ ( ) m = I ( zz ) + I ( yy ) + I ( yy ) + m + m φφφ ( ) m = I ( xy ) + m a ( z ) + m a ( z ) φφφ ( ) m = I ( yz ) φφφ ( ) m = I ( zx ) φφφ ( ) m = m a ( x ) + m + m φφφ ( ) m = m a ( y ) − m a ( z ) − m a ( z ) φφφ ( ) m = I ( xx ) − I ( yy ) − m φφφ ( ) m = I ( zz ) + m φφφ ( ) m = I ( xy ) φφφ ( ) m = I ( yz ) φφφ ( ) m = I ( zx ) − m a ( z ) φφφ ( ) m = m a ( x ) + m φφφ ( ) m = m a ( y ) φφφ ( ) m = I ( xx ) − I ( yy ) φφφ ( ) m = I ( zz ) φφφ ( ) m = I ( xy ) φφφ ( ) m = I ( yz ) φφφ ( ) m = I ( zx ) φφ ( ) m = m a ( x ) φφφ ( ) m = m a ( y ) φφφ ( ) m = I ( xx ) − I ( yy ) + I ( yy ) + I ( yy ) − m φφφ ( ) m = I ( zz ) + I ( yy ) + I ( yy ) + m + m φφφ ( ) m = I ( xy ) + m a ( z ) + m a ( z ) φφφ ( ) m = I ( yz ) φφφ ( ) m = I ( zx ) φφφ ( ) m = m a ( x ) + m + m φφφ ( ) m = m a ( y ) − m a ( z ) − m a ( z ) φφφ ( ) m = I ( xx ) − I ( yy ) − m φφφ ( ) m = I ( zz ) + m φφφ ( ) m = I ( xy ) φφφ ( ) m = I ( yz ) φφφ ( ) m = I ( zx ) − m a ( z ) φφφ ( ) m = m a ( x ) + m φφφ ( ) m = m a ( y ) φφφ ( ) m = I ( xx ) − I ( yy ) φφφ ( ) m = I ( zz ) φφφ ( ) m = I ( xy ) φφφ ( ) m = I ( yz ) φφφ ( ) m = I ( zx ) φφφ ( ) m = m a ( x ) φφφ ( ) m = m a ( y ) where M = Õ i = m i B. Rotation Matrices
The rotation matrix of the base link ( I R ) is formulated using the Z-X-Y euler angle notation. The link framesare assigned to the links using the modified DH notation [34]. Hence, for a parent-child pair i , j such that ρ j = i , the34otation matrix is: i R j = Cq j − Sq j Sq j Cq j C α j − S α j Sq j S α j Cq j S α j C α j (43)where, C • = cos (•) , S • = sin (•) . C. Computed Kinematic Data
Computing the regressor matrix requires the kinematic data, i.e., base pose, base twist, joint angle, and joint velocity.However, only the rates are assumed to be measured and the pose is computed from the rates. Further, the base angularvelocity in the inertial frame is computed from the euler rates. The quantities to be computed and measured is a choicethat can be made by the user based on the available sensors. Finally, all the kinematic data is required to construct theregressor matrix. An time index ‘ t ’ is appended to the subscript of the kinematic data.Base position and orientation at time index ‘ t ’ are as follows: r , t = r , t − + v , t − ∆ t (44) ζζζ t = ζζζ t − + Û ζζζ t − ∆ t (45)where ζζζ is a 3 × t ’ is as follows: ωωω , t = I R , t M t Û ζζζ t (46)where, M t transforms the euler rates to body rates.Eq. 44 and 45 togther provide the base pose. Measured base velocity and the computed angular velocity in Eq. 46provide the base twist.Joint angle of j th joint at time index ‘ t ’ is computed from its joint velocity as follows: q j , t = q j , t − + Û q j , t − ∆ t (47) Acknowledgment
This research was supported by Department of Science and Technology (DST), India INSPIRE research Grant(IFA-13 ENG-52) of the second author. The authors would like to thank J. Krishna Murthy, Robotics Research Center,International Institute of Information Technology, Hyderabad for his helpful suggestions on the early version of this35aper.
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