Virtual Forward Dynamics Models for Cartesian Robot Control
JJournal for Intelligent & Robotic Systems manuscript No. (will be inserted by the editor)
Virtual Forward Dynamics Models for CartesianRobot Control
Stefan Scherzinger · Arne Roennau · Rüdiger Dillmann
Received: date / Accepted: date
Abstract
In industrial context, admittance control represents an important schemein programming robots for interaction tasks with their environments. Those robotsusually implement high-gain disturbance rejection on joint-level and hide direct ac-cess to the actuators behind velocity or position controlled interfaces. Using wristforce-torque sensors to add compliance to these systems, force-resolved controllaws must map the control signals from Cartesian space to joint motion. Althoughforward dynamics algorithms would perfectly fit to that task description, their ap-plication to Cartesian robot control is not well researched. This paper proposes ageneral concept of virtual forward dynamics models for Cartesian robot control andinvestigates how the forward mapping behaves in comparison to well-establishedalternatives. Through decreasing the virtual system’s link masses in comparisonto the end effector, the virtual system becomes linear in the operational spacedynamics. Experiments focus on stability and manipulability, particularly in sin-gular configurations. Our results show that through this trick, forward dynamicscan combine both benefits of the Jacobian inverse and the Jacobian transpose and,in this regard, outperforms the Damped Least Squares method.
Keywords
Forward dynamics · robot control · kinematics In robotics, task space control is important for many applications, since it providesa natural way for programmers to specify goals and constraints. The accordingcontrol laws can be formulated in operational space of the end-effector. Since therobots are articulated mechanisms and are powered in their joints, these controllers
Stefan ScherzingerArne RoennauRüdiger DillmannFZI Forschungszentrum InformatikHaid-und-Neu-Str 10-1476131 Karlsruhe, GermanyTel.: +0049-721-9654-226E-mail: {scherzinger, roennau, dillmann}@fzi.de a r X i v : . [ c s . R O ] S e p Stefan Scherzinger et al. need to map the Cartesian control signals to the robots’ configurations space, i.e.the motor actuators. We will refer to matrices that accomplish this as mappingmatrices . Two frequently-used variants of mapping matrices are the transpose ofthe manipulator’s Jacobian and its inverse.The Jacobian transpose is an important part in many classes of control schemesfor torque-actuated robots, such as in hybrid force/motion control [1], [2], [3], par-allel force/motion control [4], and impedance control [5], [6]. In principal, theapproaches use the Jacobian transpose as static relationship between end-effectorwrenches and joint torques for controlling robots in contact with their environ-ments. Although not strictly required, control performance is generally improvedthrough decoupling robot dynamics in operational space [3], prior to mappingthe signals to joint space. In addition, there are also algorithmic solutions usingthe principle of inverse dynamics to compute suitable joint torques from motioncontrol signals, e.g. with the Recursive Newton-Euler Algorithm (RNEA) [7].An important subset of robots, however, does not provide joint interfaces ontorque level. These systems are often found in industrial context, and are the pri-mary focus of this paper. In [6], those systems are referred to as simplified systems because they hide internal dynamics decoupling behind a velocity interface. In thispaper, we will refer to them as velocity-actuated systems to underline the type ofinterface exposed by the robot vendors. On these systems, velocity-resolved controlvariants, such as admittance control [6], usually leverage Jacobian inverse-relatedmethods, such as the Damped Least Squares (DLS) [8] as the mapping to jointspace.Unlike inverse dynamics for torque-actuated systems, literature on velocity-actuated robots mostly neglects forward dynamics as an algorithmic option forcontrol. This is surprising, because it represents a straightforward mapping fromCartesian wrench space to joint accelerations. While we used this method to con-trol robots in previous work [9], [10], the new contribution of this paper is anin-depth analysis of particular features of this mapping and an evaluation againstother well-established methods. The goal and novelty is a drop-in-replacement forthe Jacobian inverse and DLS in controllers for velocity-actuated robots. Throughusing a dynamics-conditioned, virtual forward model, we match the linear, decou-pled behavior of the Jacobian inverse while simultaneously keeping the inherentrobustness in singularities of the Jacobian transpose method.The paper is structured as follows: In 2 we briefly recapitulate the inverse kine-matics mapping problem along with established methods to make it easy for thereader to follow comparisons in the experiments. Section 3 then presents forwarddynamics-based mappings for Cartesian control. In the experiments section 4, weinvestigate ill conditioning, stability and manipulability in singular configurationsand evaluate our approach against suitable subsets of the Jacobian Inverse, theJacobian transpose and the DLS method. We finally discuss remaining points andsuggestions in 5 and conclude with directions for further research in 6.
The goal of this paper is to evaluate forward dynamics-based mappings againstwell-established methods. We use Singular Value Decomposition (SVD) as a toolto investigate the characteristics of the mapping matrices. SVD factorizes a matrix irtual Forward Dynamics Models for Cartesian Robot Control 3 M according to M = U ΣV T , with σ i = Σ ii . (1) U and V T are orthogonal matrices. The entries σ i ≥ of the diagonal matrix Σ are known as the singular values and determine the scale of the mapping. For ourexperiments, σ min and σ max as the minimal and maximal singular values are ofparticular interest in analyzing stability and manipulability.To recapture some basic concepts, let the forward kinematics mapping be givenwith x = g ( q ) , (2)which computes an end-effector pose, denoted here with x from the joint statevector q . The velocity vector of generalized coordinates ˙ q maps with ˙ x = J ˙ q (3)to end effector velocity vector ˙ x , using the manipulator Jacobian J = J ( q ) . Wegenerally omit the joint vector dependency in further notation for brevity reasons.For non-redundant manipulators, the inverse mapping is given by ˙ q = J − ˙ x . (4)Near singular configurations, J looses rank, such that its inverse becomes numeri-cally unstable. The respective mapping for end-effector forces and torques to jointspace with τ = J T f (5)does not suffer from these instabilities. However, J becoming rank-deficient meansthat some components of f will lie in the nullspace of the Jacobian transpose, i.e.they will be balanced by the mechanism’s mechanics and will hence not be able toactuate the joints. This effect is a severe limitation for controller implementations.Applied to motion control, a formal investigation of the Jacobian transposemethod and a numerical solution to the Inverse Kinematics problem was presentedin [11]. The authors’ solution derives from a simple, 2nd order dynamical system ¨ q = KJ T (cid:16) x d − g ( q ) (cid:17) , (6)computing joint accelerations from the difference of a desired pose x d and thecurrent pose as determined by the forward kinematics g ( q ) . They show with aLyapunov stability analysis that the system is asymptotically stable for an arbi-trary positive definite matrix K .Using J T will serve as a lower bound and baseline in stability considerationsof our contribution.The DLS method is an applications of the Levenberg-Marquardt stabilizationto manipulator control [12], [8] and tries to remove instabilities of J − near singular configurations. Note that the original version as proposed in [8] uses partial velocitymatrices, adding the benefit of allowing different reference frames for each element.Since we don’t make use of this feature, we replace it with the common manipulatorJacobian J instead. Similar to pseudo inverse methods for redundant systems,which minimize (cid:107) J ˙ q − ˙ x (cid:107) , the idea is to add a damping term α against excessive Stefan Scherzinger et al.
Fig. 1
Illustration of pulling an exemplary robot manipulator into singularity. joint velocities that will trade-off accuracy for stability near singular configurationswith (cid:107) J ˙ q − ˙ x (cid:107) + α (cid:107) ˙ q (cid:107) . (7)The solution that minimizes this quantity is given by ˙ q = ( J T J + α I ) − J T ˙ x , (8)see e.g. [13] for a derivation. The matrix ( J T J + α I ) is non-singular, which canbe shown with singular value decomposition [13] and hence is guaranteed to beinvertible. This method is well established for practical control implementationsand can serve as a drop-in replacement for J − in control loops. We use thismethod as a baseline to compare our new forward dynamics-based method interms of manipulability.A popular enhancement to DLS, using this method, is Selectively DampedLeast Squares (SDLS) [14]. The method converges faster and circumvents to choosea suitable α by introducing singular vector-specific damping terms of the singularvalue decomposition of J at the expense of a higher runtime cost.Other methods include the more recent Exponentially Damped Least Squares(EDLS) [15], which is a solution with the focus on physical Human-Robot interac-tion (pHRI). Although effectively avoiding elbow-lock and wrist-lock among othercommon singular phenomena, it requires explicit, albeit easy parameterization bythe user.Both the Jacobian inverse and the Jacobian transpose have strengths and short-comings and present mappings for physically different control spaces. When usedon velocity-actuated systems, the Jacobian inverse does not need dynamic decou-pling, but suffers from instability, which DLS effectively mitigates at the expense ofloosing accuracy with increased damping. The Jacobian transpose needs dynamicsdecoupling in the controller, but offers inherent stability near singular configura-tions. A general incentive is to obtain the best of these corner cases. This paperproposes and empirically evaluates forward dynamics as a suitable approach toachieve this combination at the core of closed-loop control schemes. f is pulling irtual Forward Dynamics Models for Cartesian Robot Control 5 the mechanism into singularity, in which the mechanism yields the external forcesas good as it can, limited by kinematic constraints. In the fully stretched case,increasing f will not create further motion. The robot’s mechanics compensatethe external load until a possible breakage of the links. This behavior is in factinverse to how J − would compute joint motion due to an external error vector,where theoretically infinite joint velocity would occur.To make use of forward dynamics simulations, robotic manipulators can bemodeled as a system of articulated, rigid bodies. The equations of motion describethe relationship between generalized loads τ in the joints, external loads f , actingon the end-effector and motion in generalized coordinates q with the followingordinary differential equations in symbolic matrix notation τ + J T f = H ( q ) ¨ q + C ( q , ˙ q ) + G ( q ) . (9) H denotes the mechanism’s positive definite inertia matrix, C comprises the Cori-olis and centrifugal terms and G is the vector of gravitational components. Forwarddynamics computation has the goal of solving Eq. (9) for q ( t ) , i.e. simulating themechanism’s reaction motion through time, given external loads.Literature has proposed several methods for forward dynamics computations,which can be categorized [7] as mainly belonging to inertia matrix methods withimplementations of the composite rigid body algorithm , e.g. in [16], [17], or thepropagation methods with the articulated body algorithm [18] being an importantrepresentative. We refer the interested reader to [7] for an broad coverage of thefield and recent implementation of various algorithms.Forward dynamics is a substantial component in multi body simulations. Thefact that it is, however, mainly neglected for closed-loop control on velocity-actuated systems may stem from the fact that computing good approximationsfor H and C of the robots is extremely difficult. The required crucial data, suchas link masses and inertia tensors is hardly available in data sheets. On the otherhand, a second reason for not being used might be that even if those data wereavailable, the benefit of forward simulating highly realistic motion would get lostwhen executed on velocity-actuated systems. Their internal joint servos with high-gain disturbance rejection could not make use of the accuracy of dynamics thatwas used to generate that motion. The reference trajectory to follow would appearas any arbitrary trajectory.This thought points to an interesting opportunity: We could reduce Eq. (9) toa rough simplification and investigate, if it’s possible to condition H to beneficially tweak the behavior of this mapping when using it as a forward model in closed-loopcontrol.3.2 A general closed-loop controlTo motivate simplifications to Eq. (9), we investigate how a controller would per-ceive the system in a possible closed loop control. A general scheme is shown in Fig. 2. A suitable control law computes a Cartesian control signal f c , using a userspecified reference input and a controlled variable as feedback from the robot. Notethe role of the virtual system as a forward model in the controller: We simulate howour proxy system behaves and send that as a reference to the real system. Sincewe obtain joint accelerations as a response from our forward model, we integrate Stefan Scherzinger et al.
RobotmanipulatorControllaw ReferencesignalReferenceinput Virtualsystem ControlledvariableControlsignal
Fig. 2
Closed-loop control with forward dynamics. We use a virtual system to forward simulatejoint motion as a reference to the real robot. those signals before sending them as reference to the real robot. The advantageis, that this virtual model will react kinematically and mechanically plausible toexternal loads f c , as was illustrated in Fig. 1.From the control law’s point of view, a linear, virtual system would be beneficialfor using constant control gains for wide regions of the robot’s joint configurationspace. By dropping the gravity term ( G ( q ) = ) from Eq. (9), we assure that thecontrol law does not need to constantly compensate this virtual load. If we furtherconsider instantaneous motion for each control cycle, i.e. accelerate from rest with ˙ q = , we can drop the non-linearities C ( q , ˙ q ) and obtain ¨ q = H − ( q ) J T f c (10)as an unbiased forward mapping. We also set τ ≡ to emphasize that f c shall bethe only virtual load guiding the virtual system.While dropping these terms reduces computational complexity in our con-troller, including them can offer additional configuration. This is briefly discussedin section 5.Note that H − ( q ) needs to be computed in each control cycle due to its depen-dency of the current joint state. In the experiments section, we evaluate computa-tional cost in comparison to other mapping matrices. Since Eq. (10) is effectivelya Jacobian transpose-based method, the next step is to decouple our virtual H − .3.3 Decoupling virtual dynamicsWe start with the time derivative of Eq. (4) ¨ x = ˙ J ˙ q + J ¨ q (11)and consider instantaneous accelerations in each cycle while the virtual system isstill at rest, so that ˙ J ˙ q = . With Eq. (10) we obtain ¨ x = JH − J T f c = Λ − f c , (12)which describes the Cartesian instantaneous acceleration of the virtual system dueto the Cartesian control input f c . The quantity Λ is known as the mass matrix in operational space, see e.g. [3], [6], with Λ = J − T HJ − .The intention of our dynamic decoupling is to make Λ − a time-invariant, di-agonal matrix across joint configurations q . This ideal mapping is illustrated inFig. 3. In order to preserve consistency of our virtual system and the real robot, weuse identical kinematics for both systems. This ensures that the reference signals irtual Forward Dynamics Models for Cartesian Robot Control 7 Fig. 3
Graphical illustration of the mapping from wrench space to Cartesian acceleration. Λ is a × matrix for both redundant and non-redundant systems. Our goal is to obtain adecoupled, diagonal mapping for arbitrary joint configurations. Fig. 4
Dynamics-conditioned, virtual model for an exemplary robot. The goal is to make themechanism behave as a unit mass, which is illustrated with the oversized sphere. for the real robot to follow agree with possible limits. We are, however, free inchanging the dynamics of the virtual H to obtain the desired effect, in particularits mass distribution. The Cartesian control signal f c acts directly on the virtualmechanism’s end effector. If that end-effector link is dominant with respect to theoverall dynamics, determined by the other links, we could obtain a behavior thatcomes close to an idealized unit mass. Fig. 4 illustrates this phenomenon. As a con-sequence, the overall systems’ center of mass roughly stays with the end-effector.Likewise does the operational space inertia Λ depend less on joint configurations,and f c experiences the same rotational inertia for both configurations. Havinga realistic link mass distribution would instead mean higher inertia with greaterdistance to the rotary axis. To measure the effect of end-effector mass dominance,we define γ = m e m l = ip e ip l (13)to be the ratio of end-effector mass m e and link mass m l . The quantities ip e and ip l denote the polar momentums of inertia of the end-effector and the other links,respectively. In the experiments section, we empirically show that increasing γ indeed leads to the desired behavior and provides a decoupled virtual system forCartesian closed-loop control. f c = x d − g ( q ) .In [11], the authors prove with a Lyapunov stability analysis that a closed loopsystem, built from this mapping, is asymptotically stable for an arbitrary positive Stefan Scherzinger et al.
DLSFDJIJTAbbr.
Fig. 5
Mapping matrices for the experiments. The abbreviations stand for Jacobian inverse(JI), Jacobian transpose (JT), Damped Least Squares (DLS) and Forward Dynamics (FD).Two types are investigated: (a) Mappings from Cartesian space to joint space and (b) mappingsfrom Cartesian space to Cartesian space. definite matrix K . This formal proof also includes our proposed H − , which is, dueto being grounded in the manipulator’s kinetic energy T = ˙ q T H ˙ q , also positivedefinite. We evaluated our forward dynamics-based approach against the DLS and against the two corner cases J − and J T in various experiments. We chose the UniversalRobot UR10’s kinematics for our experiments. Our perception is that this robotis well-known and used both in industry and academia and therefore presents asuitable platform.Depending on the phenomena investigated, a subset of different mapping ma-trices was used. An overview of these matrices and their composition is given inFig. 5 along with the abbreviations used in the plots. We implemented each map-ping matrix literally, i.e. as a multiplication of the respective symbols in C++,using the robot’s kinematics from a popular ROS [19] package and the algorithmsfor computing J and H from a well-established robotics library .For all experiments, the following values were chosen for the forward dynamicsmappings: m e = 1 kg , m l = m e γ , ip e = 1 kg / m , ip l = ip e γ (14)The ratio γ was then varied according to the investigation of each experiment.4.1 DecouplingIn this experiment, we evaluated the effectiveness of our virtual model dynam-ics decoupling and compared the mapping to both the Jacobian inverse and the Jacobian transpose as reference. The mapping matrices were of type (b) fromFig. 5. Fig. 6 shows the results of the analysis. All mean matrices are diagonal,which is to expect for sampling a vast amount of arbitrary joint configurations. https://github.com/ros-industrial/universal_robot https://github.com/orocos/orocos_kinematics_dynamicsirtual Forward Dynamics Models for Cartesian Robot Control 9 M e a n JI JT FD γ =1 FD γ =10 e S t d . d e v i a t i o n . . . . . . . . . . Fig. 6
Analysis of the × mapping matrices of type (b) from Fig. 5. The plot shows theindividual matrix entries in form of 2d heat plots. To obtain the plots, we sampled . random configurations of q uniformly with q i ∈ [ − π, π ] and computed the respective mappingmatrix for each type. The figures show mean and standard deviation for the entirety of samples. The standard deviations, however, show a strong configuration dependency for theJacobian transpose. This mapping would be suboptimal if used in a closed-loopcontrol scheme. Instead, the Jacobian inverse behaves ideal and converges to theidentity matrix. Note that using forward dynamics with an even mass distribution( γ = 1 ) already improves upon the Jacobian transpose. The experiment furthershows, that with a significant end-effector mass dominance ( γ = 10 e ) the forwarddynamics mapping converges to the Jacobian inverse and makes this mappingparticularly suitable for closed-loop control in terms of linearity.4.2 Ill-conditioned configurationsIn this experiment, we continued the evaluation of decoupling and compared FDand DLS with regard to ill conditioning of the mapping matrices from Fig. 5(b).Higher numbers of ill conditioning degrade control performance [20], but heavilydepend on the manipulators configuration. This experiment investigates how FDand DLS influence ill conditioning by varying γ and α , respectively. Based on[20], we used κ = σ max /σ min as the measure for ill conditioning. Fig. 7 shows theresults. For each discrete point in the plots, we evaluated random joint stateswith the limits from Fig. 6. We used quartiles on our data to effectively excludeoutliers ( σ min → ), such that the plots show the median of the ill conditioning. Itcan be seen that FD converges much faster to beneficial condition numbers overits own parameter space than DLS. In fact, most of the decoupling effect fromexperiment Fig. 6 is already available for low values of γ . J γ ˜ κ = σ m a x σ m i n FD ( γ )0 . . . . . . α DLS ( α ) Fig. 7
Ill conditioning κ for the DLS and FD method. Each data point is the median of randomly evaluated joint configurations . . . σ m i n J − J T steps σ m a x Fig. 8
The two baselines J T and J − while passing through four singular configurations,illustrated with dashed, vertical lines. becomes rank-deficient. This is an unfortunate joint configurations for all consid-ered approaches. As a consequence, the manipulator is not able to achieve in-stantaneous motion in all directions [21]. Two issues arise from this constellation:First, Jacobian transpose-based methods tend to loose manipulability. We measurethis effect with σ min of the mapping matrix, which is one of various establishedmeasures [21]. Second, for Jacobian inverse-based methods, infinite joint velocityoccurs. We measure this effect with σ max as an indicator of how much the mappingmatrix scales f c in sensitive dimensions to joint space. The goal of the experimentis to investigate how well each approach behaves in singular configurations con-cerning both measures. As a reference, Fig. 8 shows both the Jacobian inverse andthe Jacobian transpose for a pass through four singular configurations, the first two being close together.The curves show the expected and well-known effect: The Jacobian inversemaintains high manipulability at the cost of an exploding σ max , while the Jacobiantranspose stays stable throughout the pass but cannot avoid σ min dropping to zeroin singularities. irtual Forward Dynamics Models for Cartesian Robot Control 11 steps . . . . . . σ m i n J − J T DLS α =0 . FD γ =25 FD γ =250 FD γ =500 FD γ =1000 Fig. 9
Investigation of σ min of various mapping matrices through four singular configurations. steps σ m a x J − J T DLS α =0 . FD γ =25 FD γ =250 FD γ =500 FD γ =1000 Fig. 10
Investigation of σ max of various mapping matrices through four singular configura-tions. Fig. 9 and Fig. 10 finally show the behavior of forward dynamics with a setof different γ . The curves show how FD approaches the Jacobian inverse whilemaintaining σ max in stable ranges. We added the DLS method, albeit with onlyone α , for comparison. Both FD and DLS have very similar characteristics andshow a good trade-off between both corner cases (JI, and JT). Note how the curvesfor FD become more pronounced towards the Jacobian inverse for increasing γ .4.4 Empirical analysis of stability and manipulabilityIn this experiment, we wanted to analyze FD in comparison to DLS on a broaderscale. The goal is an empirical analysis of varying α (DLS) and γ (FD) over biggerranges, and evaluate how they perform in the corner cases in comparison to theJacobian inverse and transpose. Instead of focusing a few trajectories, we sampleda massive amount of singular constellations. Note that in contrast to the workspace sampling for experiment 4.1, which contained singular configurations by chance,in this investigation we exclusively used singular configurations. Through exclu-sively focusing regions of low performance (singularities) throughout the wholeworkspace, the results become a feasible measure of global performance for eachmethod. γ . . . . . σ m i n ( . ) σ m i n ( J − ) FD ( γ )0 . . . . . . α DLS ( α ) Fig. 11
Relative manipulability for the DLS and FD method in comparison to J − . Note,that practical applications of the DLS method may require higher damping values of up to α = 1 . as reported in [14] To find a big amount of singular configurations, we first sampled equally dis-tributed, random joint states as start configurations. We then used Particle SwarmOptimization (PSO) [22], which implements an adapted algorithm from the orig-inal work of [23] as a black-box optimizing strategy to converge to singular con-figurations from these start states. We used Yoshikawa’s manipulability measure (cid:112) det ( JJ T ) , which simplifies for non-redundant mechanisms to | det ( J ) | [24] asfunction to minimize, which was faster than using SVD with σ min directly. Alter-natively, a more type-based approach of finding singularities is discussed in [25],[26].Having a set of singular configurations, we then computed average valuesfor σ min and σ max from the mapping matrices of FD and DLS according to Fig. 5(a)for discrete values of α and γ for each of the singular configurations. Fig. 11shows the results for manipulability. Both DLS and FD approach the Jacobianinverses behavior for decreasing α and increasing γ , respectively. Note how FDapproaches qualitatively faster in its own parameter space.Fig. 12 shows the results for stability. DLS comes closer to the Jacobian trans-poses stability than FD throughout most of the observed parameter space. How-ever, towards reaching the Jacobian inverses high manipulability, DLS looses sta-bility and asymptotically approaches infinity, while FD in contrast stays bounded.For applications in which DLS would require very low values of α for control per-formance, FD can be used as a safe alternative, combining and keeping the benefitsof both J − and J T .4.5 Computational efficiency Finally, we measured average execution times of the mapping approaches. Weimplemented the SDLS method according to [14] and included the measurementsas additional reference.To obtain the comparison, we computed ¨ q as given in Fig. 5(a) with a ficti-tious, constant f c = for e times. The joint state q was randomly sampled, irtual Forward Dynamics Models for Cartesian Robot Control 13 γ σ m a x ( . ) σ m a x ( J T ) FD ( γ ) 0 . . . . . . α DLS ( α ) Fig. 12
Relative instability for the DLS and FD method in comparison to J T . . . . . . . . µs JTJIDLSFDSDLS
Fig. 13
Execution times of computing different mappings of type (a) from Fig. 5 on an Intel R (cid:13) Core TM i7-4900MQ. while being identical across one single evaluation of each method. Fig. 13 showsthe boxplots of each method’s execution time with their quartiles. The medianis plotted as vertical, orange line. The whiskers for minimal an maximal execu-tion times indicate a high degree of irregularity. We expect narrower ranges forexperiments on a hard real-time operating system.The results show that the forward dynamics method is a little more computa-tionally intense than the DLS method, but approximately half the execution timeof the SDLS. Being still in the low µ s range makes forward dynamics in the versionfrom this paper suitable for real-time closed-loop control. For redundant manipulators, including those terms offers additional interfaces toadjust behavior in the nullspace of the Jacobian transpose. For those cases, switch-ing from the composite rigid body algorithm to propagation methods for solvingthe forward dynamics might be beneficial. The articulated body algorithm [18],e.g. allows an intuitive integration of external loads to each link of the robot sepa- rately, which might be used to implement collision avoidance or other optimizationsconcerning the robots’ posture.5.2 Control applicationsThe natural mapping of forward dynamics from Cartesian wrench space to jointaccelerations makes it particularly suitable for the implementation of admittance-related controllers on velocity-actuated systems. For those use cases, force-resolvedcontrol laws for disturbance rejection can replace the velocity-resolved control lawsusing DLS. The benefit of using the FD method is its ability to operate extremelyclose to the ideal J − behavior without significantly sacrificing stability. Successfulimplementations of forward dynamics-based control on industrial robots can befound e.g. in [27] for pure force control and in [9], [28] for compliance control. Anapplication to motion control with a particular focus on sparsely sampled targetsis presented in [10]. This paper proposed virtual forward dynamics models for Cartesian robot control.The core of the control loop is a simplified, virtual model that maps Cartesiancontrol signals to joint accelerations. Through increasing the end effector’s massin comparison to the other links, the virtual system becomes linear in the opera-tional space dynamics and matches the exactness of the Jacobian inverse. Furtherexperiments showed, that this forward model’s decoupling leads to less ill condi-tioning compared to the DLS method for an empirical investigation of the jointspace. When passing through singularities, forward dynamics behaves in generalsimilar to DLS in terms of manipulability and stability. Yet, when operating insingular configurations forward dynamics models substantially differ from DLSin that they produce bounded control signals, even when forced to approach theJacobian inverse in terms of manipulability. These virtual forward models are par-ticularly suitable for implementing admittance controllers in industrial settings onvelocity-actuated robots. Their computation time in the low µ s range makes themsuitable for real-time control. References [1] Daniel E Whitney. “Force feedback control of manipulator fine motions”. In:(1977).[2] Marc H Raibert and John J Craig. “Hybrid position/force control of manipu-lators”. In:
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