A Cable-Driven Parallel Robot with Full-Circle End-Effector Rotations
Marceau Métillon, Philippe Cardou, Kévin Subrin, Camilo Charron, Stéphane Caro
AA Cable-Driven Parallel Robot with Full-CircleEnd-Effector Rotations
Marceau M ´etillon , Philippe Cardou , K ´evin Subrin , Camilo Charron , St ´ephane Caro ∗ Laboratoire des Sciences du Num ´erique de Nantes (LS2N), UMR CNRS 6004, 44300 Nantes, France Centre National de la Recherche Scientifique (CNRS), 44321 Nantes, France Laboratoire de robotique, D ´epartement de g ´enie m ´ecanique, Universit ´e Laval, Qu ´ebec, QC, Canada Univ ´ersit ´e de Nantes, IUT, 44470 Carquefou, France ´Ecole Centrale de Nantes, 44321 Nantes, FranceEmails: [email protected], [email protected],[email protected], [email protected], [email protected] Cable-Driven Parallel Robots (CDPRs) offer high payloadcapacities, large translational workspace and high dynamicperformances. The rigid base frame of the CDPR is con-nected in parallel to the moving platform using cables. How-ever, their orientation workspace is usually limited due to ca-ble/cable and cable/moving platform collisions. This paperdeals with the design, modelling and prototyping of a hybridrobot. This robot, which is composed of a CDPR mountedin series with a Parallel Spherical Wrist (PSW), has both alarge translational workspace and an unlimited orientationworkspace. It should be noted that the six degrees of free-dom (DOF) motions of the moving platform of the CDPR,namely, the base of the PSW, and the three-DOF motion ofthe PSW are actuated by means of eight actuators fixed to thebase. As a consequence, the overall system is underactuatedand its total mass and inertia in motion is reduced.
A Cable-Driven Parallel Robot (CDPR) belongs to a par-ticular class of parallel robots where a moving platform islinked to a base frame using cables. Motors are mounted ona rigid base frame and drive winches. Cable coiled on thesewinches are routed through exit points located on the rigidframe to anchor points on the moving platform. The pose(position and orientation) of the moving platform is deter-mined by controlling the cable lengths.CDPRs have several advantages compared to classi-cal parallel robots. They are inexpensive and can coverlarge workspaces [1]. The lightweight cables contributeto the lower inertia of the moving platform and conse-quently to a better dynamic performance over classical par-allel robots [2]. Another characteristic of the CDPRs is theirreconfigurability. Changing the overall geometry of the robotcan be done by changing the exit points and anchor points. ∗ Address all correspondence to this author.
Reconfigurability of the CDPRs is suitable for versatileapplications especially in an industrial context [3, 4]. CD-PRs have drawn researchers’ interests towards robotic appli-cations such as pick-and-place operations, robotic machin-ing, manipulation, intralogistics measurements and calibra-tion systems [5, 6].CDPRs can offer an extremely large three-degreesof freedom translational workspace, but their orientationworkspace is usually limited due to cable/cable and ca-ble/moving platform collisions [7]. Cable interferences canbe divided into cable-cable and cable-environment interfer-ences [8]. For a given pose of the moving platform, cable-cable collision may occur by changing the moving platformorientation. Cable-environment collisions refer to the inter-ferences between the moving platform and its surroundingenvironment.This paper presents the concept of a hybrid manipula-tor with decoupled translation and orientation motions of themoving platform. It is called hybrid since two parallel mech-anisms are connected in series [9], namely, a CDPR anda Parallel Spherical Wrist (PSW). In [10], a hybrid CDPRequipped with multiple platforms and up to nine cables isable to orientate a end-effector around an axis with an unlim-ited range of rotation. In [11,12], redundant drive wire mech-anisms are used for producing motions with high accelera-tion and good precision. In [13,14,15], differential cables areused to increase the size of the manipulator workspace. Dif-ferential cables consists into a set of two independent cablesconnecting the platform to two different winches, the latterbeing actuated by a differential mechanism driven by a sin-gle actuator. In [16], two spring-loaded cable-loops allowsfor the control of a two-DOF planar CDPR with only two ac-tuators. Loaded springs ensure the compliance given by thevariation of cable lengths and grants a higher stiffness to themechanism. In [17], a bi-actuated cable is used in order toimprove the orientation capacity of the end-effector of pla-nar cable-driven robots. In [18] a hybrid CDPR uses a cable-1 a r X i v : . [ c s . R O ] J a n A A A A A A A F (cid:2)x (cid:2)y (cid:2)z O rigid frame motor gearbox winchcables moving platformexit points top plateSphere(End Effector) armcablesanchor pointsspring doubleomni-wheel F (cid:2)x (cid:2)y (cid:2)z O B B B B B B B B (cid:2)x (cid:2)y (cid:2)z O F Fig. 1: CDPR with full-circle end-effector rotationsloop for remote actuation of an embedded hoist mechanismon the moving platform. In [19], a moving-platform embed-ding a two-DOF differential gear set mechanism is presentedallowing the two-DOF unlimited rotational motions of theend-effector. Thanks to cable-loops (bi-actuated cables), theembedded mechanisms are actuated by transmitting powerthrough cables from motors, which are fixed on the ground,to the moving platform [20]. Therefore, by remote actuationthrough cable-loops, the tethering of the power cable to themoving platform is eliminated [21]. Moreover, a lower massand a lower inertia of the moving platform are obtained dueto the remote actuators.In this paper the mechanical design of the manipulatorunder study is presented in Section 2. Its kineto-static modelis described in Section 3. The static workspace of the manip-ulator is studied in Section 4. The methodology followed todetermine the optimal cable arrangement of the system is ex-plained in Section 5. The developed prototype of the mech-anism at hand and some experimental results are shown inSection 6. Conclusions and future work are drawn in Sec-tion 7.
The kinematic architecture of the manipulator consistsof two parallel manipulators mounted in series. A CDPRgrants a large translation workspace and a PSW grants anunlimited orientation workspace. This hybrid manipulator isable to combine advantages of both mechanisms in terms oflarge translation and orientation workspaces.
Figure 1 shows the overall architecture of the manipula-tor with its main components, namely, winches, exit-pointsand the moving platform. The winches control the cable lengths, which move and actuate the moving platform. Ca-bles are routed through exit-points located on the rigid frameand connected to anchor-points located on the moving plat-form.Figure 1 shows the moving platform, which hosts a topplate assembly and embeds the PSW. The end-effector of thewrist is a sphere actuated by three cable-loops, which trans-mit the required power from motors fixed on the ground tothe end-effector of the moving platform, namely the sphere.The PSW is described in detail in Sec. 2.2 while thecable-loop system is presented in Sec. 2.3. The top platehas the six-DOF and the PSW grants a large orientationworkspace to the sphere providing an overall nine-DOFworkspace to the spherical end-effector regarding the baseframe F . The concept of the PSW relies on the Atlas platformprinciple [22]. An end-effector is linked to the base of thewrist using a spherical joint. Three omni-wheels are linkedto the wrist base with revolute joints. The end-effector is asphere actuated by the rotation of the omni-wheels. The posi-tion of the wheel relative to the sphere surface was defined toallow a singular-free and unlimited orientation of the spherearound the global (cid:126) x , (cid:126) y and (cid:126) z axes as discussed in [23].Here, the top plate of the CDPR amounts to the wristbase. Three carriage sub-assemblies are rigidly attachedto the top plate. Every carriage hosts an omni-wheel. Athree rigid arm platform hosts the sphere using three casterballs. Three anti-backlash compression springs are mountedon threaded rods using nuts, thus ensuring the connection ofthe arm platform to the top plate. The springs ensure an ad-justable contact force of the omni-wheels on the sphere. Theomni-wheels transmit torque to the sphere thanks to friction.2ach omni-wheel is independently driven by a cable-loopsystem. Figure 2 represents a simplified form of the cable-loopsystem. The latter consists of a single cable of which bothends are actuated by two motors while passing through exit-points, namely, A , A , and anchor-points, namely, B , B .The cable-loop is coiled up around a drum on the movingplatform. The cable-loop drum then acquires one rotationalDOF with respect to the moving platform. These can be usedto actuate an embedded mechanism or to control additionaldegrees of freedom such as rotations over wide ranges. Thepurpose of the cable-loop is double. Firstly, its aim is totranslate the top plate as two single cables would do whenthe coiling directions of both motors are the same. Secondly,it actuates the embedded drum by circulating the cable whencoiling directions of the two actuators are different. motor motor base top plate t ωτ t drum rB B A A Fig. 2: Representation of a cable-loopBy controlling the difference of tension in both ends ofthe cable loop, namely δ t = t − t , it is possible to transmittorque τ to the pulley. This capacity can be used to increasethe orientation workspace of the end-effector when the pul-ley is used to rotate the end-effector or the pulley can actuatean embedded mechanism without having the drawbacks ofembedding the actuators on the platform. Here, three ca-ble loops are used to actuate independently the three omni-wheels in contact with the sphere as illustrated in Fig. 2. As shown in Fig. 1, F denotes the frame fixed to thebase of origin point O and axes (cid:126) x , (cid:126) y and (cid:126) z . F is theframe attached to the top-plate of origin point O and axes (cid:126) x , (cid:126) y and (cid:126) z . F is the frame attached to the end-effector,i.e., the sphere, of origin point O , the geometric center ofthe sphere and axes (cid:126) x , (cid:126) y and (cid:126) z .It is noteworthy that F and F have a translational andorientational relative movement while F and F only have arelative rotational movement.The exit points A i are the points belonging to the framethrough which the cables are routed between the winch and the top plate. The anchor points B i are the points belongingto the TP where the cables are connected. It is noteworthythat the cables are connecting exit points and anchor pointsaccordingly and a unit vector expresses the cable direction.Therefore, the loop-closure equations associated with eachcable are expressed as follows: l i = a i − p − R b i (1)with i ∈ [[ , . . . , ]] where l i is the i -th cable vector, a i is thecorresponding anchor point expressed in the base frame, b i is the coordinate vector of exit point in the platform frame, p is the position vector of the platform frame and R is therotation matrix from F to F .We can then write the i -th unit cable vector as: u i = l i l i (2)with l i being the i -th cable length. In this section, we proceed to the kinetostatic modellingof the overall manipulator. We write the static equation ofthe manipulator as follows: Wt + w e = (3)with W being the wrench matrix, t being the cable tensionvector and w e being the external wrenches applied on theplatform. In our case, we only consider the action of theweight of the moving platform as external wrench.The wrench matrix of the manipulator W is the concate-nation of the wrench matrices of both mechanisms: W = (cid:20) W TP W SW (cid:21) × (4)where W TP is the wrench matrix associated to the top plateand W SW is the wrench matrix related to the PSW.Similarly, the wrench vector of the manipulator consistsof the wrench vector of both mechanisms: w g = (cid:20) w TPg w SWg (cid:21) × (5)where w TPg is the wrench vector associated to the top plateand w SWg is the wrench vector exerted on the PSW. W TP and w TPg are defined in Section 3.1 while W SW and w SWg are de-fined in Section 3.2.3 .1 Kinetostatic Model of the Top Plate
In this section we write the kinetostatic model of the topplate. The static equilibrium of the platform can be writtenas: W TP t + w TPg = (6) W TP takes the following form: W TP = (cid:20) u u u u u u u u d d d d d d d d (cid:21) × (7)with d i being the cross-product of vectors b i and u i ex-pressed in the base frame F as: d i = R b i × u i (8)We define the wrench vector of the top plate w TP as follows: w TPg = (cid:20) f TP m TP (cid:21) × (9)where f TP is the 3-dimensional force and m TP is the 3-dimensional moment exerted on the top plate. The parametrization of the PSW is illustrated in Fig. 3and described below: C i : Contact point between the i th omni-wheel and thesphere Π i : Plane passing through the contact point C i and tangentto the sphere α : Elevation angle of point C i , α ∈ [ , π ] β : Angle between the tangent line L i and the actuation forceof the omni-wheel, β ∈ [ − π , π ] γ i : Angle between (cid:126) x and the vector pointing from point H to point C i , i = , , r s : Sphere radius r o : Omni-wheel radius˙ ϕ i : Angular velocity of the i -th omni-wheel v i : Unit vector along the actuation force produced by the i -th omni-wheel on the sphere n i : Unit vector normal to plane Π i The optimal set of the parameters of the wrist was de-fined in [23] in order to maximize the amplitudes of its ori-entation as well as its dexterity. The following hypothesesare taken into account for the wrist modelling and analysis:( i ) The omni-wheels are normal to the sphere; ( ii ) The con-tact points between the omni-wheels and the sphere belong to the circumference of a circle. The latter is the base of aninverted cone, its tip being the centre of the sphere. The an-gle between the vertical axis and the cone is named α ; ( iii ) Inthe plane containing the cone base, the three contact pointsform an equilateral triangle. r s r o O H C i n i ϕ i α βγ i (cid:6)x (cid:6)y (cid:6)z Π i L i v i omni-wheelsphere Fig. 3: Parametrization of the Parallel Spherical WristThe angular velocity vector of the sphere ω =[ ω x , ω y , ω z ] T is expressed as a function of the angular veloc-ity vector of the omni-wheels ˙ ϕ = [ ˙ ϕ , ˙ ϕ , ˙ ϕ ] T , as follows: A ω = B ˙ ϕ (10) A and B are the forward and inverse Jacobian matrices of thePSW, which take the form: A = r s ( n × v ) T ( n × v ) T ( n × v ) T (11)and, B = r o × (12)From Eq. (11), matrix A is expressed as a function ofangles α , β and γ i as: A = r s S β C γ − C α C β C γ − S β C γ − C α C β C γ S α C β S β C γ − C α C β C γ − S β C γ − C α C β C γ S α C β S β C γ − C α C β C γ − S β C γ − C α C β C γ S α C β (13)4here C k = cos ( k ) and S k = sin ( k ) , k = α , β , γ , γ , γ .From [23], the PSW is fully isotropic from a kinematic view-point if and only if α = . ◦ and β = ◦ . Therefore, thosevalues are selected in what remains. Equation (10) is rewrit-ten as follows: ω = J ω ˙ ϕ (14)where J ω = A − B , is the Jacobian matrix of the wrist, i.e., J ω is the mapping from angular velocities of the omni-wheelsinto the required angular velocity of the end-effector. Basedon the theory of reciprocal screws [24], it turns out that: m T SW ω = τ T ˙ ϕ (15)where m SW = [ m x , m y , m z ] T is the output moment vectorof the sphere and τ = [ τ , τ , τ ] T is the input torque vec-tor, namely the omni-wheel torque vector. By substitutingEq. (14) into Eq. (15), we obtain: τ = J T ω m SW = W ω m SW (16)The wrench matrix W ω = J T ω maps the output torque ofthe sphere to the omni-wheels torques. As shown in Fig. 4, τ can be expressed as a function of the cable tensions suchthat: τ = r d ( t − t ) (17) τ = r d ( t − t ) (18) τ = r d ( t − t ) (19)where r d is the radius of the embedded drum of the cable-loops.From Eqs. (17) to (19), the omni-wheel torque vector τ is expressed as a function of the cable tensions as follows: τ = W c t (20)where t = [ t , . . . , t ] T is the cable tension vector and W c is the matrix assigning the cables to the cable-loops. If cablepairs (1, 2), (3, 4) and (5, 6) respectively correspond to cable-loops 1,´e 2 and 3, W c will take the form: W c = r d − r d r d − r d r d − r d (21)The equilibrium of the wrench applied on the PSW isexpressed as: m SW = W SW t (22) with m SW being the external moments applied by the envi-ronment onto the PSW. The wrench matrix W SW expressesthe relationship between the cable tensions and the momentapplied on the wrist. W SW = W ω W c (23)The orientation of the sphere with respect to the baseframe is defined by the pitch angle θ , the yaw angle ψ andthe roll angle χ while following the ZYX-Euler-angles con-vention. Those three angles are the components of the orien-tation vector q SW of the wrist, namely, q SW = θψχ (24) t i + t i − r d ω i τ i Fig. 4: Representation of a cable-loop drum
The static workspace of the manipulator consists of theset of positions and orientations of the moving platform andthe orientations of the end-effector, namely, p and R and q SW , which satisfies the static equilibrium of the manipula-tor. The cable tension set T amounts to a hyper-cube in aneight-dimensional space: T = { t ∈ R : t min ≤ t ≤ t max } (25)where t min and t max are respectively the lower and upperbounds of the cable tension.The static workspace of the manipulator is defined asfollows:5 = { ( p , R , q SW ∈ R × SO ( ) × R : ∃ t ∈ T , Wt + w e = } (26)where SO ( ) is the group of proper rotation matrices. Thestatic workspace in a nine-dimensional space is expressedusing the Equation (26). As the visualization of such high-dimensional space is impossible with common human per-ception in 3D, we define the static workspace of the manipu-lator for a simplified case. We define S for a constant orien-tation of the top-plate. The former subset, namely, S AO is aset for a given orientation of the top-plate while the wrist isfree to rotate: S AO = { p ∈ R | R = I | − π ≤ θ , ψ , χ ≤ π : ∃ t ∈ T , Wt + w e = } (27)The discretization of the Cartesian space is made so that n x , n y and n z are the numbers of discretized points along (cid:126) x , (cid:126) y and (cid:126) z axes, respectively. R S is defined as the proportion of the static workspaceto the overall space occupied by the manipulator: R S = N S ( n x + )( n y + )( n z + ) (28)with N S being the number of points inside the discretizedstatic workspace S . This section deals with the cable arrangement of themanipulator for obtaining the maximum size of the staticworkspace. The i -th cable arrangement is the association ofanchor points B i to exit points A i . The top plate has fifteenpoints for anchor points and the manipulator has a maximumof eight actuators. Therefore, the optimal cable arrangementis defined as the association of eight anchor points to eightexit points such that the workspace is maximized.The number of exit point combinations, N e is: N e = (cid:18) n e n c (cid:19) (29)with n e and n c being the numbers of available exit-points andcables, respectively. The number of anchor-points combina-tions, N a , consists in the number of permutations of the setof points, which is given by: R R R R R R R R R R R R R R R F O (cid:2)x (cid:2)y Fig. 5: The fifteen points for the anchor points on themoving-platform N a = (cid:18) n a n c (cid:19) n c ! (30) n a being the number of selected anchor points. S C is the setof possible cable configuration, the number of cable config-uration N C = dim ( S C ) is thus given by: N c = N a N e = (cid:18) n e n c (cid:19)(cid:18) n a n c (cid:19) n c ! (31)Figure 5 shows all the available anchor points,namely, r i , i = , , . . . ,
15, which are divided intotwo groups: S CL = { R , R , R , R , R , R } which isthe set of points associated to cable-loops and S SC = { R , R , R , R , R , R , R , R , R } which is the set ofpoints associated to the simple cables. Six points amongstthe fifteen points are selected to make the three cable loops.Therefore, nine remaining anchor points host the remainingtwo single-actuated cables. n c = n SC + n CL (32)The number of available anchor points for the single andbi-actuated cables are denoted as n aSC and n aCL :6 a = n aSC + n aCL (33)The number, N CL , of combinations considering cable-loop is given by: N CL = (cid:18) n e n c (cid:19)(cid:18) n aSC n SC (cid:19) n c ! (34)Six points are associated to the wrist actuation and con-sequently to the three cable-loops: n aCL =
6. The remain-ing nine anchor points are to be assigned to simple cables n aSC =
9. Finally, N CL , expressed in Eq. (34) is given by: N CL = (cid:18) (cid:19)(cid:18) (cid:19) = N CL is very large, for computing-time sake, the num-ber of available points for anchor points is supposed to beequal to three. Thus S SC = { R , R , R } . By substituting n aSC = N CL = (cid:18) (cid:19)(cid:18) (cid:19) =
120 960 (36)The static workspace is computed for the 120 960 ca-ble arrangements. Figure 6 illustrates the cable arrangementcorresponding to the largest workspace. Figure 6a showsa schematic of the anchor-plate as well as the optimal ca-ble arrangement. Figure 6b shows the corresponding staticworkspace with R S = The prototyping of a CDPR with full-circle end-effectorrotations is presented in this section. The base frame ofthe prototype shown in Fig. 7 is 4 m long, 3.5 m wide and4 m high. The full-circle end-effector rotations are obtainedthanks to the PSW shown in Fig. 8. The PSW is mainly madeup of a top plate, three double omni-wheels and a transpar-ent sphere. The three-DOF rotational motions of the sphereare obtained from the rotations of the omni-wheels. Eachomni-wheel is driven by a cable loop, the strands of the ca-ble being respectively wound around two actuated reels fixedto the ground. A flight controller © Pixhawk, embedded inthe sphere, is used to measure its orientation, angular veloc-ity and linear acceleration. This controller, equipped witha gyroscope, an accelerometer and a magnetometer, acts asa data-logger device to record measurements. The overallmass of the PSW is equal to 1.87 kg. It should be noted x [m] y [m] . .
85 1 . .
95 22 . . . . B ≡ R B ≡ R B ≡ R B ≡ R B ≡ R B ≡ R B ≡ R B ≡ R B B B B B B B B F O (cid:2)x (cid:2)y (a) Cable configuration x [m] y [m] z [m] 0 . . .
50 0 01 1 12 2 23 34 (b) Static workspace
Fig. 6: Optimum cable arrangement and static workspaceFig. 7: The CDPR prototype with full-circle end-effector ro-tations7 op platespherePixhawk omni-wheelbattery
Fig. 8: The parallel spherical wrist equipped with a Pixhawkflight controllerthat the manipulator is under-actuated because the moving-platform including the PSW has nine degrees of freedomwhereas the prototype has only eight actuators.Figure 9 shows the main hardware of the prototype,which consists of a PC (equipped with © MATLAB and © ControlDesk software), eight © PARKER SME60 motorsand TPD-M drivers, a © dSPACE DS1007-based real-timecontroller and eight custom-made winches.Slippage between the omni-wheels and the sphereinevitably leads to drift in the orientation when tracking aprescribed trajectory. Therefore, this robot is intended for EthernetBi-directionnalcommunicationReal-time controllerMotor driver Cable lengthCurrent setpointEncoder position Motor, gearbox and winchSupervision PC
Fig. 9: Equivalent architecture of the prototype teleoperation applications where an operator can compensatefor the errors that accumulate over time. In this context, therobot performance is best assessed by comparing its angularvelocity response to an angular velocity input rather than bylooking at its ability to track position and orientation over atrajectory.The performances of the protoype were experimentallyevaluated along the following three trajectories:
Trajectory 1:
Pure rotational motions of the sphere aboutaxes parallel to x , y and z , respectively, while the topplate is fixed to its support. A fifth-order polynomialis used to obtain continuous velocity and accelerationtrajectory profiles. Trajectory 2:
Pure translational motions of the top platealong four successive straight line segments whereas thesphere does not rotate. A fifth-order polynomial is usedto determine the velocity and acceleration profiles alongeach line segment.
Trajectory 3:
The top plate performs a vertical translationalmotion while the sphere rotates. t [s] ω x [ d e g / s ] DesiredMeasured (a) about x axis t [s] ω y [ d e g / s ] DesiredMeasured (b) about y axis t [s] ω z [ d e g / s ] DesiredMeasured (c) about z axis Fig. 10: Desired and measured angular velocities of thesphere along Trajectory 1A video illustrates the experiments carried out with theprototype. The experimental motions of the sphere alongTrajectory 1 are shown in the video from 0min 5s to 1min 7s. https://bit.ly/3hvpsL2 ω x ( ω y , ω z , resp.) aboutaxis x ( y , z , resp.) of the sphere expressed in frame F . x y z ω [ d e g / s ] MaxRMS
Fig. 11: Maximum and root mean square errors on the sphereangular velocity along Trajectory 1As shown in Fig. 11, the maximum error on the sphereangular velocity along Trajectory 1 is about 30 deg/s. Itshould be noted that the root-mean square error of the an-gular velocity of the sphere is smaller about x than about z .Although errors are present between the desired and obtainedangular velocities of the end-effector, those experiments con-firm the fidelity of the PSW kinematic model obtained inEq. (23), the kinematic Jacobian matrix of the cable-actuatedPSW being the opposite of the transpose of the wrench ma-trix W SW . The errors between measured and desired end-effector angular velocities are mainly due to the slippage be-tween the omni-wheels and the sphere.The experimental motions of the wrist along Trajec-tory 2 are shown in the video from 1min 7s to 3min 41s.Let θ e denote the rotation angle of the top plate. θ e is de-fined from R as follows: θ e = arccos (cid:18) tr ( R ) − (cid:19) , ≤ θ e ≤ π (37)Figure 12 shows the evolution of θ e , i.e., the parasiticinclination of the top plate, along the four straight line seg-ments of Trajectory 2. Figure 12 illustrates the large transla-tion workspace of the manipulator as presented in Figure 6b.It is apparent that the maximum parasitic inclination angle ofthe top plate along this trajectory is about 10 deg. Further-more, Fig. 12 shows the validity of the kinematic model ofthe top plate defined in Eq. (7). The kinematic model of therobot is well validated quantitatively considering the smallvalue of the parasitic inclinations. Nevertheless, its staticmodel could not be validated other than by observing thatinclinations of the top plate are mainly due to friction in thePSW to be identified and errors in the identified Cartesiancoordinates of the cable exit points.The experimental motions of the wrist along Trajec-tory 3 are shown in the video from 3min 41s to 5min 38s. It https://bit.ly/3hvpsL2 z [ m ] y [m] x [m] θ e [ d e g ] Fig. 12: Parasitic inclination of the top plate along Trajec-tory 2is noteworthy that parasitic inclinations of the top plate ap-pear once the sphere starts rotating. These parasitic motionsare caused by several factors. Some improvements in thePSW design and robot control are needed to address theseweaknesses. First, the cable anchor points located on the topplate and shown in Fig. 5 are very close to each other lead-ing to a high sensitivity of the orientation of the top plate tovariations in cable lengths. Thus, this kinematic sensitivityshould be reduced by moving the cable anchor points awayfrom each other on the top plate. Besides, it is difficult tomanage ( i ) the friction between the sphere and the three bear-ings that support it, ( ii ) the friction and contact between theomni-wheels and the sphere, and ( iii ) the friction and slip-page between the cable loops and the three pulleys aroundwhich the cables are wrapped. Indeed, it can be noted in theprevious videos that the omni-wheels sometimes start to slideon the sphere and that they do not turn at other times whenthe cable loop is supposed to circulate. To resolve these is-sues, im t is necessary to better manage the cable tensionsand improve the manufacturing and assembly of some partssuch as the connection between the double omni-wheels andthe top plate. Finally, the flexibility of the three arms holdingthe sphere may be responsible for bad contacts between thesphere and the omni-wheels. This paper dealt with the design, modelling and proto-typing of a hybrid robot. This robot, which is composed of aCDPR mounted in series with a PSW, has both a large trans-lational workspace and an unlimited orientation workspace.It should be noted that the six degrees of freedom motions ofthe moving platform of the CDPR, namely, the base of thePSW, and the three-DOF motion of the PSW are actuated by9eans of eight actuators fixed to the base. As a consequence,the overall system is underactuated and its total mass and in-ertia in motion is reduced.The kinetostatic model of the hybrid robot was ex-pressed in order to determine its static equilibrium condi-tion and find the cable arrangement maximizing its staticworkspace. A prototype of the CDPR with full-circleend-effector rotations was realized. The performances ofthe prototype were experimentally evaluated by measuringits angular-velocity response over and parasitic inclinationsalong three test trajectories. It appears that the maximum an-gular velocity error of the sphere is about 30 deg/s along thefirst trajectory for which the top plate remains on its support.Future work will focus on reducing parasitic inclinations ofthe end-effector thanks to an improved design of the wristand better management of the cable tensions.
Acknowledgements
This work was supported by both the ANR CRAFTproject, grant ANR-18-CE10-0004 and the RFI At-lanSTIC2020 CREATOR project. Assistance provided byDr. Saman Lessanibahri and M. Julian Erskine through theexperimentation process is highly appreciated.
Nomenclature F ( O , x , y , z ) Base frame attached to the rigid frame ofthe robot F ( O , x , y , z ) Frame attached to the top plate F ( O , x , y , z ) Frame attached to the end-effector of theParallel Spherical Wrist l i i -th cable vector pointing from B i to A i , i ∈ [[ , . . . , ]] a i Cartesian coordinates vector of point A i , i ∈ [[ , . . . , ]] b i Cartesian coordinates vector of point B i , i ∈ [[ , . . . , ]] p Cartesian coordinates vector of point P u i i -th cable unit vector, i ∈ [[ , . . . , ]] t Cable tension vector w e External wrench vector w TPg
External wrench vector associated to the top plate w SWg
External wrench vector associated to the ParallelSpherical Wrist d i Cross-product of b i and u i , i ∈ [[ , . . . , ]] f TP Force applied on the top plate m TP Moment applied on the top plate m SW External moments applied by the environment ontothe end-effector v i Axis of actuation force produced by the i -th omni-wheelon the sphere n i Unit normal vector of i -th omni-wheel on plane π i atpoint C i q SW Orientation vector of the end-effector l i i -th cable length, i ∈ [[ , . . . , ]] r s Radius of the sphere r o Radius of the omni-wheels r d Radius of the embedded drum of the cable loop A i i -th cable exit point, i ∈ [[ , . . . , ]] B i i -th cable anchor point, i ∈ [[ , . . . , ]] C i Contact point between the i -th omni-wheel and thesphere, i ∈ [[ , , ]] R i Available anchor point on the top plate i ∈ [[ , . . . , ]] ˙ ϕ i Angular velocity of the i -th omni-wheel, i ∈ [[ , , ]] ω α Angle associated to the position of the contact points C i on the sphere, α ∈ [ , π ] β Angle between tangent to the sphere and the actuationforce of the omni-wheel, β ∈ [ − π , π ] γ i Angle between the contact points C i and (cid:126) x , i ∈ [[ , , ]] θ Pitch angle of the end-effector ψ Roll angle of the end-effector χ Yaw angle of the end-effector τ i i -th omni-wheel torque R Rotation matrix from frame F to frame F W Wrench matrix W TP Wrench matrix related to the top plate W SW Wrench matrix related to the Parallel Spherical Wrist A Forward Jacobian matrix of the Parallel Spherical Wrist B Inverse Jacobian matrix of the Parallel Spherical Wrist W c Wrench matrix of the cable-loop Π i Plane passing through the contact point C i and tangentto the sphere, i ∈ [[ , , ]] L i Line tangent to the sphere on C i in the plane ( (cid:126) x , (cid:126) y ) S Static workspace of the manipulator S AO Static workspace of the manipulator with a given ori-entation of the top plate R S Proportion of the static workspace to the overall spaceoccupied by the manipulator N S Number of points inside the discretized staticworkspace N e Number of exit point combinations N a Number of anchor points N C Number of cable configurations N CL Number of cable configurations considering cable-loop n e Number of available exit-points n c Number of cables n a Number of selected anchor points n aSC Number of available anchor points for the single cable n aCL Number of available anchor points for cable-loop S C Set of possible cable configurations
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