Robotica (27) volue 25, pp. 5 52. 27 Cabridge University Press doi:.7/s26357477342 Printed in the United Kingdo Coparison of null-space and inial null-space control algoriths Bojan Neec, Leon Žlajpah and Dair Orčen Robotics Laboratory, Jožef Stefan Institute, Jaova 39, Ljubljana, Slovenia (Received in Final For: January 4, 27. First published online: February 26, 27) SUMMARY This paper deals with the stability of null-space velocity control algoriths in extended operational space for redundant robots. We copare the perforance of the control algorith based on the inial null-space projection and generalized-inverse-based projection into the Jacobian nullspace. We show how the null-space projection affects the perforance of the null-space tracking algorith. The results are verified with the siulation and real ipleentation on a redundant obile robot coposed of 3 degrees of freedo (DOFs) obile platfor and 7-DOF robot ar. KEYWORDS: Redundant anipulator; Stability of the control algoriths; Autonoous otion; Obstacle avoidance; Mobile anipulator.. Introduction One of the iportant issues of the new generation of service and huanoid robots is the kineatic redundancy. The kineatic redundancy is characterized by extra degrees of freedo (DOFs) with respect to the given otion posed by the assigned priary task. Most huanoid robots, as well as robotic ars ounted on a obile platfor, are kineatically redundant. A redundant anipulator has the ability to ove the end-effector along the sae trajectory using different configurations of the echanical structure. This provides eans for solving sophisticated otion tasks such as avoiding obstacles, avoiding singularities, optiizing anipulability, iniizing joint torques, etc. The consequence is a significant increase in the dexterity of the syste, which is essential to accoplish coplex tasks. Additionally, redundancy also has an iportant influence on the dynaic behavior of the robotic syste. An appropriate control of dynaic properties is essential for higher perforance. The ajority of control algoriths proposed in the past decade are acceleration-based redundancyresolution schees, where appropriate joint accelerations are generated in oer to accoplish the priary task and null-space otion as a secondary task. The behavior of a redundant anipulator is deterined by the apping of joint velocities into the Jacobian null-space. For redundant anipulators, the end-effector dynaics is only one part of the dynaics of the whole anipulator. The rest * Corresponding author. E-ail: bojan.neec@ijs.si dynaic represents the dynaics of the internal otion of the anipulator, which has to be controlled in oer to accoplish the desired secondary task. Velocity null-space control is an appropriate way to control internal otion. It is well established that certain acceleration-based control schees exhibit instabilities, especially in instantaneous torque iniization redundancy-resolution schees. 2,22 These instabilities were recently atheatically analyzed by O Neil. 23 It has also been deonstrated that energydissipation controllers, 7 which can be interpreted as velocity null-space controllers with zero desired null-space velocity, can becoe unstable with low-velocity feedback gain. 23 The sae result was published previously, 5 where Lyapunov functions were used to deterine the range of stability of the velocity controller. We 6 proposed a odification that assures the stability of the controller for low gains of the null-space controller. Siilar result was reported earlier. 4 Another approach based on dynaic decoposition of kineatically redundant anipulators into the task-space dynaics and null-space dynaics based on a inially reparaetrized hoogenous velocity was proposed by Chang 2 and later by Park et al. 24 and Oh et al. 2 It has been shown earlier 25 that this approach does not exhibit instabilities, not even with instantaneous torque-iniization redundancy resolution. This paper copares approaches based on generalizedinverse-based redundancy resolution at velocity level and inial null-space redundancy resolution at velocity level. In both cases, we copared ipedance force controller in extended task-space 2 as a ore general control. We have highlighted the conceptual differences between both approaches. Furtherore, we analyze stability of algoriths based on a inially paraeterized null-space projection atrix subjected to the calculation of null-space projection atrix. We deonstrate the perforance degradation of the control schee due to the non-unique representation of nullspace projection atrix. We solved this proble related to the real ipleentation by applying an appropriate ethod for calculation of the null-space projection atrix. 2. Kineatics Robotic systes under study are n-dof serial anipulators. We consider only redundant systes, which have ore DOF than needed to accoplish the task, i.e. when the diension of the joint-space n exceeds the diension of the task-space, n>and denote r = n as the degree of redundancy. Let the configuration of the anipulator be represented
52 Coparison of null-space and inial null-space control algoriths by the vector q of n joint positions, and the end-effector position (and orientation) by -diensional vector x of task positions (and orientations). The relation between joint and task velocities is given by the following well-known expression: ẋ = J q () where J is the n anipulator Jacobian atrix. The solution of the above equation for q can be given as a su of particular and hoogeneous solution where q = q p + q h = Jẋ + Nξ (2) J = W J T (JW J T ) (3) where J is the weighted generalized-inverse of J, W is the weighting atrix, N = (I JJ) is n n atrix representing the projection into the null-space of J, and ξ is an arbitrary n-diensional vector. We will denote this solution as the generalized-inverse-based redundancy resolution at the velocity level. 9 The hoogenous part of the solution belongs to the Jacobian null-space. Therefore, we will denote it as q n. Since the rank of the null-space atrix N is r, the hoogeneous solution in Eq. (2) can also be presented in the for q n = Nξ = Vẋ n (4) where V is a full colun rank n r atrix which satisfies the criteria JV =, and ẋ n is an arbitrary r-diensional inial null-space velocity vector. This approach will be denoted as inial null-space redundancy resolution at velocity level. 3. Generalized-Inverse-Based Velocity Control (NSVC) First we will derive the control law using generalized-inversebased redundancy resolution at velocity level in the extended operational space. We denote this approach as null-space velocity control (NSVC). Extended operational space is an extension of the operational spaces introduced by Khatib 6 and offers unique approach for the analysis of both task-space and null-space. 2,2 Let us define extended-space variable x e as [ ] [ ] ẋ J ẋ e = = J e q = q (5) ξ N where J e is the extended Jacobian. Since q = Jẋ + Nξ, the generalized inverse of J e is defined as [ ] J e = J N (6) The proposed selection of the generalized inverse J e satisfies the generalized inversion property J e J e J e = J e (7) and inversion property J e J e = JJ + NN = JJ + I JJ = I (8) The anipulator dynaics in joint-space is described by τ = H q + h + J T F (9) where H is n n inertia atrix of the anipulator, h is n-diensional vector of centrifugal, coriolis, and gravity forces, and F is n-diensional vector of external forces acting on the anipulator s end effector. Preultiplying Eq. (9) by J T e and using q = J e ẋ e, the equation of otion can be reforulated using the extended task-space variables as where and f e = e ẍ e + µ e + F e () f e = J e τ () e = J T e H J e = J T H J J T HN (2) J T HN N T HN µ e = J T e h e J e q (3) F e = [ ] F (4) A straightforwa approach to the controller design is to copensate nonlinear coupling ters of the syste dynaics and apply a control vector, which assures the desired syste dynaics. Let us define the control force in extended operational space as f c = e ẍ c + µ e + F e (5) where ẍ c denotes the control vector in the for ẍ c = ẍd + K v ė x + K p e x (6) q nd + K n ė n Here, e x = x d x is the task-space tracking error and ė n = q nd q n is the null-space tracking error. x d and q nd are the desired task cooinates and null-space velocity, respectively. Inserting Eq. (5) into () yields e (ẍ c ẍ e ) = (7) The general for of the extended operational space inertia atrix e contains off-diagonal ters, which eans that the task-space and the null-space are inertially coupled. By selecting W = H in Eq. (3), we obtain J T HN = (JH J T ) JH H(I H J T (JH J T ) J = (JH J T ) J (JH J T ) J = (8)
Coparison of null-space and inial null-space control algoriths 53 This shows that the inertia-weighted generalized inverse is the only one that decouples task-space and null-space otion 4. Equation (7) is thus decoupled into two equations J T H J(ë x + K v ė x + K p e x ) = N T HN(ë n + K n ė n ) = (9) Since atrix J T H J is positive definite, it follows that ë x + K v ė x + K p e x =. In contrast, the atrix N T HN is only positive sei-definite and this does not iply that ë n + K n ė n =. Using identity N T HN = HN and a positive definitness of inertia atrix H yields N(ë n + K n ė n ) =. Therefore, null-space projection of the null-space tracking error tends to zero, but this does not iply that null-space tracking error itself tends to zero. This is also the cause of instabilities of acceleration-based redundancy-resolution schees which use a velocity null-space controller. Naely, O Neil 23 proved that the energy-dissipation schee proposed by Khatib 7 cannot guarantee the stability at low gains of the dissipation energy controller. Siilar results were obtained earlier 5 by using the Lyapunov stability criteriu. By setting the desired null-space velocity to zero, the null-space velocity controller reduces to an energy-dissipation controller. The joint-space control law can be derived by inserting Eq. (6) into Eq. (5) and preultiplying by J e τ c = H J(ẍ d + K v ė x + K p e x J q) + HN( q nd + K n ė n Ṅ q) + h + J T F (2) The first ter corresponds to the task-space control τ x, the second to the null-space control τ n, and the thi and the fourth to the copensation of the nonlinear syste dynaics and external force, respectively. Using the identity NṄ = N JJ, q nd = N q d + Ṅ q d and introducing joint-space error e q = q d q, the null-space control law can be rewritten as τ n = HN( q d + K n ė q + Ṅė q ) = HN( q d + K n ė q JJė q ) (2) 4. Minial Null-Space-Based Velocity Control (MNSVC) The transforation fro joint cooinates to inial nullspace velocities is described by ẋ n = V q (22) where V is the generalized inverse of V and is defined as V = (V T WV) V T W (23) where W is the weighting atrix. Using the above forulation, we can define the extended-space x e as ẋ e = [ ẋ ẋ n ] [ ] J = J e q = q (24) V where J e is the extended Jacobian. The sybol is used to denote that the corresponding variable is defined in inial null-space in contrast to the variables without the suffix, which are defined in generalized-inverse-based null-space. Since q = Jẋ + Vẋ n, the inverse of J e is defined as [ J e = J V] (25) The null-space atrix N and inial null-space atrix V are related through N = V V (26) The above relation relies on the definition of V, and is easily verified by inserting Eq. (22) into Eq. (2). Siilar to the previous case, the equation of otion can be reforulated using the extended task-space variables where and f e = eẍ e + µ e + F e (27) f e = J e τ (28) e = J e H J e = J T H J J T HV V T H J V T HV (29) µ e = J T e h e J e q (3) F e = [ ] F (3) Let us define control vector in extended-space in the sae way we did it in generalized-inverse-based controller. Inserting Eq. (32) into (27) yields f c = eẍ c + µ e + F e (32) e(ẍ c ẍ e) = (33) where ẍ c denotes the control vector in the for ẍ c = ẍd + K v ė x + K p e x (34) ẍ nd + K n ė n Variable ė n = ẋ nd ẋ n denotes the velocity tracking error in inial null-space. Again, by selecting W = H in Eq. (23), the off-diagonal eleents of the extended inertia atrix are zero. Equation (34) is thus decoupled into two equations J T H J(ë + K v ė x + K p e x ) = V T HV(ë n + K n ė n) = (35) Since both atrices J T H J and V T HV are positive definite, it follows ë x + K v ė x + K p e x = and ë n + K n ė n =. The
54 Coparison of null-space and inial null-space control algoriths error equation shows the ain advantage of the inial null-space approach. Only inial null-space approach assures the desired dynaic behavior in the null-space, which cannot be guaranteed for the generalized-inversebased controller. The reason for this is in the existence of the generalized inverse of inial null-space transforation atrix V. On the contrary, atrix N is rank deficient and inverse of N is singular. It was proved by eans of O Neil identity that inial null-space acceleration redundancyresolution schees are not subjected to torque instabilities. 25 However, in the next section, we will show that perforance degradation of the control algorith arises due to the nonunique representation of null-space projection atrix. Again, the joint-space control law can be siplified to τ c = H J(ẍ d + K v ė x + K p e x J q) + HV(ẍ nd + K n ė n V q) + h + J T F (36) The first ter corresponds to the task-space control τ x, the second to the null-space control τ n, and the thi and the fourth to the copensation of nonlinear syste dynaics and external force, respectively. Coparing Eqs. (2) and (36), we notice that MNSVC and NSVC algoriths differ only in null-space control. Let us rewrite second ter of Eq. (36) using ẋ n = V q and Eq. (26) into τ n = H(N( q d + K n ė q ) + V Vė q ) (37) Fro the above equation, it is evident that the only difference between NSVC and MNSVC is how they copensate the projection of the joint-space error to the null-space due to the configuration change. NSVC uses ter NṄė q, while MNSVC copensate this effect with the ter V Vė q. 5. Minial Null-Space Calculation There is an infinite nuber of possible null-space transforations. We have shown 7 that the null-space otion depends only on the criteria function and the selected weighting atrix. Therefore, it is independent of the selection of the nullspace transforation V. On the other hand, the nuerical stability of the control algorith is subjected to the selection of V. Naely, representation of the null-space with the base vectors is not unique. There is an infinite nuber of orthonoral basis vectors V that describe the sae null-space. For good control, it is necessary to obtain a sooth continuous solution of V during the execution of the robot s task. There are several ethods for obtaining V. The ethod proposed by Park et al. 24 uses singular value decoposition (SVD) of J, or alternatively, J T J. Singular value decoposition or J yields U Z T = J (38) where is the diagonal n-diensional atrix of nonzero eigenvalues denoted by s and n zero eigenvalues of J. [ ] R The corresponding atrices Z and U have for Z = V and U T = [Q V T ]. Since atrices U and Z are unitary, it follows and [ [ ] R Q V T J V U T JZ = ] = s s2... s (39) Obviously, QJV =, V T JR =, and since Q and R are nonzero atrices, JV =, V T J = and thus sub-atrix V fors null-space of J. Unfortunately, atrix V is not unique. There is an infinite nuber of orthonoral basis vectors V that describe the sae null-space. The ost popular technique for coputing the SVD is the Golub Reunsch algorith and is available in any linear algebra software packages. It is regaed as the ost efficient and nuerically stable technique for coputing the SVD of an arbitrary atrix. Unfortunately, it does not assure continuous solutions. For exaple, if atrix J changes continuously, this does not iply that atrices U,, and Z will also change continuously. We will deonstrate this effect with the siulation of 5 instances of the kineatics of the 4-DOF planar anipulator with links of equal length. The V atrix was calculated using Matlab function null, which is based on SVD calculation using Golub Reunsch algorith. The null-space velocity x n = V[,,, ] was applied to initiate null-space otion. At q = [2.8274, 2.3229,.477,.284], V suddenly changes the set of values. If we reverse the null-space otion, V changes again. This is shown in Fig.. Discontinuity of eleents in V causes perforance degradation of the control algorith. Naely, control algorith 36 requires V,.5.5 Eleents of V 2 3 4 5 saples (t) 3 2 joint cooinates 2 3 4 5 saples (t) Fig.. Eleents of V and joint angles q.
Coparison of null-space and inial null-space control algoriths 55 which has to be differentiated nuerically. Discontinuity of V results in an unbounded control signal. We solved the above proble using SVD algorith based on Givens rotations. 8 The approach was reviewed by Maciejewski and Klein as an algorith, ore suited to take advantage of increental perturbations and parallel architectures. For our purpose, we do not need to calculate all atrices of SVD. We need only the atrix Z, which orthogonalizes the coluns of J. This atrix is usually fored as a product of successive Givens rotations, each orthogonalizes two coluns. Considering the current ith and jth coluns of J, a ultiplication by Givens rotation results in new coluns J i = J i cos(θ) + J j sin(θ) J j = J j cos(θ) J i sin(θ) (4) with constraint J i J j =. The ters in the Givens rotation atrix which ortogonalizes J can be coputed by using the following forulas:,3 p = J T i J j (4) q = J T i J i J T j J j (42) v = 4p 2 + q 2 (43) For q, the rotation atrix eleents are v + q cos(θ) = 2v p sin(θ) = v cos(θ) (44) (45) and for q<, we can use another set of eleents in oer to avoid ill-condition. v q sin(θ) = sig(p) (46) 2v p cos(θ) = (47) v sin(θ) However, orthogonalization cannot be achieved in single sweep. In general, we need ultiple sweeps, but the algorith converges.,3 Perhaps, the ost useful property of the algorith is the ability to use perturbed initial values of atrix Z. The ore orthogonal are the coluns of JZ, the fewer are the nuber of sweeps required for the convergence, and even ore iportant in our case, the solutions are continuous. If one considers the current J to be a perturbation of the previous J, J(t + δt) = J(t) + δj(t), then the atrix J(t + δt)z(t) will have nearly orthogonal coluns. Since control of the anipulator consists of subsequent calculations of Eq. (36), we can use the solution of Z fro the previous step, which iproves the convergence of the algorith, reduces coputational buen, and assures contiguous solution of the Jacobian null-space atrix V, which is ost iportant. In real ipleentation, special care should be paid to the orthogonality test of coluns J. Nor.5.5 Eleents of V 2 3 4 5 saples (t) 3 2 joint cooinates 2 3 4 5 saples (t) Fig. 2. Eleents of V and joint angles q. is not a good easure because J T i J j could be sall siply, because of sall eigenvalues. Nash 3 has proposed nor J T i J j (J T i J i )(J T j J j ). Unfortunately, when the denoinator is equal to zero, Eq. (45) is also singular. In such a case, we have found an appropriate solution by perturbation of the Jacobian with sall rando values. For the illustration, we repeated previous experient again, this tie using the proposed algorith. In Fig. 2, we can observe sooth transition of eleents of V. 6. Null-Space Motion Generation The desired extended null-space velocities, which iniize the given criteria p, can be obtained using the weighted gradient optiization procedure,5 q n = NH p q k o (48) which assures the best optiization step in the case of inertia weighted generalized inverse. k o is a negative constant and defines the optiization step. The force and the position tracking are usually of the highest priority for a force-controlled robot. The selection of the subtasks with lower priority depends on the specific application. However, collision avoidance is of great iportance in ost applications of redundant robot systes, since it is very difficult to predict the path of all links. In ost cases, the otion is not guaranteed to be conservative. Therefore, one collision-free task cycle does not iply next collision-free cycle. Following the idea of the obstacle avoidance using the potential field, 5 we define the cost function p = 2 Ed2, where E is an l l rotation atrix describing the direction of an artificial potential field pointing fro the obstacle, l is the diension of the position sub-space, and d is the shortest distance between obstacle and the robot body. In our case, the desired objective is fulfilled, if the iaginary force is applied only on robot joints. In this case, we can obtain cost function
56 Coparison of null-space and inial null-space control algoriths gradient in siple for as p q = (d J, + d 2 J,2 + +d n J,n )E (49) where d i is the vector of the shortest distances between the ith joint and the obstacle and J,i denotes the Jacobian atrices between the base (the first index in the superscript) and the ith joint (the second index in the superscript) regaing the robot positions only. In our experients, we will also use singularity avoidance. A suitable easure for deterining vicinity of singular point was proposed by Yoshikawa 27 and is described by p s = JJ T (5) Unfortunately, partial derivation q JT required by the gradient optiization (48) is generally not easy to calculate in an analytical way. Therefore, we have used nuerical derivative in our experients. 7. Experients The experiental setup consists of 7-DOF robot ar Mitsubishi PA, ounted on the holonoous obile platfor Noad XR4 with 3 DOFs. The entire setup is presented in Fig. 3. The robot ar is torque controlled using ArcNet protocol. Unfortunately, the obile platfor has no torque input, and can only be velocity controlled. Therefore, a control algorith for this syste has to be odified. 2 Unfortunately, this odification does not allow to copletely decouple task-space and null-space dynaics, and both algoriths have virtually equal response. Therefore, we copared siulation results of both algoriths. Siulation was accoplished in Matlab/Siuling and accurate dynaic odels were developed using SDFast tool. The priary task of the anipulator was to track the line. The desired speed was.45 /s and the initial joint configuration of the robot was [.5,,,,π/2,, π/2,,, ]. There was an obstacle in the robot work-space, as shown in Fig. 3. Service robot during the task. J Fig. 3. The secondary subtask was obstacle avoidance and singularity avoidance. Regaing the given task, the degree of redundancy was 7. A high degree of redundancy was selected in oer to verify that the atrix V is continuous and liited during the otion. On the other hand, a high degree of redundancy cobined with obile platfor requires careful selection of the secondary tasks. Tasks such as anipulability optiization or torque optiization will always lock the robot ar into the optial position and the task otion will be perfored with obile base only. In oer to avoid such a situation, singularity avoidance algorith was activated only if the robot was close enough to the singular configuration. The siulation results of the given task using NSVC and MNSVC control algoriths are presented in Figs. 4 and 5, respectively. The task tracking error, the null-space velocity error, and the desired null-space velocity as a result of the optiization procedure. As expected, task-space errors are identical and practically equal to zero in both cases. The null-space tracking error increases at the oent when singularity avoidance, and later, obstacle avoidance generate null-space otion. We can see that, although the null-space tracking error is saller in the case of MNSVC, there is no significant difference in robot otion in the case when NSVC and MNSVC algorith is used. This is because the null-space velocity feedback gain K n stabilizes null-space control loop. When coparing null-space tracking errors of both algoriths, we ust be aware that null-space errors are not presented in the sae space, and therefore, direct coparison is not possible. We also copared the perforance of both algoriths on a real robot. As we entioned before, the obile platfor has no torque input. Therefore, we ipleented both algoriths only on PA robot ar. We used inertia paraeters, daping and friction paraeters as published previously. 8,26 The sapling tie in our experients was.2 s. Again, the task of the robot was to track a line with speed.2 /s. In this experient, only positions were considered in the priary task. Therefore, the degree of redundancy was 4. In oer to copare the perforance of both algoriths equally, we explicitly generated the secondary otion. The secondary otion was sinusoidal velocity with aplitude.5 rad/s and frequency.6 s, applied to joints 6 and 7 of the anipulator. The trajectory was copleted in 5 s, then we left the robot to perfor only the secondary otion for another 5 s. The priary task tracking errors, null-space tracking errors, and copensation signals Ṅ q and V q for the NSVC and the MNSVC control algoriths are presented in Figs. 6 and 7, respectively. As expected, task errors are siilar in both cases, while the null-space tracking errors are significantly lower with the MNSVC algorith. As we entioned previously, it is difficult to copare null-space errors of both algoriths, since they are presented in different null-spaces. Therefore, in figures we projected inial null-space errors into the generalized-inverse-based null-space, using the equation ė n = Vė n. Fro the results, it can be clearly seen that the NSVC algorith does not fully copensates the influence of the joint otion, which is, accoing to Eqs. (2) and (37), the only difference between both algoriths. We also copared the desired and the obtained secondary velocity.
Coparison of null-space and inial null-space control algoriths 57 5 TASK ERROR.5.5 2 2.5 3 3.5 4 NS ERROR.5.5 2 2.5 3 3.5 4 5 OPTIMIZATION SIGNAL /s Fig. 4. Siulation results with NSVC. 5.5.5 2 2.5 3 3.5 4 5 TASK ERROR.5.5 2 2.5 3 3.5 4 MNS ERROR.5.5 2 2.5 3 3.5 4 5 OPTIMIZATION SIGNAL /s 5.5.5 2 2.5 3 3.5 4 Fig. 5. Siulation results with MNSVC.
58 Coparison of null-space and inial null-space control algoriths 5 4 TASK ERROR 5 2 4 6 8. NS ERROR. 2 4 6 8 2 3 COMPENSATION SIGNAL Fig. 6. Experiental results with NSVC. 2 2 4 6 8 5 4 TASK ERROR 5 2 4 6 8. MNS ERROR. 2 4 6 8 4 COMPENSATION SIGNAL 2 2 Fig. 7. Experiental results with MNSVC. 4 2 4 6 8 The results are presented in Fig. 8 and are alost identical for both algoriths. Although null-space tracking errors are alost zero, the control algoriths alost perfectly track the desired secondary otion of joint 7, while tracking of joint 6 is iperfect. The reason is that the otion of joint 7 is already in the null-space and does not affect the priary task.
Coparison of null-space and inial null-space control algoriths 59 /s /s.5 DESIRED AND OBTAINED q6 velocity.5 2 4 6 8.5 DESIRED AND OBTAINED q7 velocity.5 2 4 6 8 Fig. 8. Experiental results with MNSVC. On the contrary, the otion of joint 6 also affects the priary task. The apping to the null-space changes its aplitude. We can also see that the obtained otion is configurationdependent. This siple exaple shows that perfect tracking of the secondary task is not very iportant, as the apping into the null-space changes the desired secondary otion. More significant is, therefore, the closed loop stability of the algorith. Null-space control strategies without nullspace velocity feedback exhibit instabilities if generalizedinverse-based apping of the Jacobian to the null-space is used instead of inial null-space transforation. We repeated the sae experient with the MNSVC control also, by using the Golub Reunsch algorith for the SVD calculation. The results are presented in Fig. 9. We obtained siilar results as with Givens-rotations-based inial nullspace calculation. However, due to the discontinuous changes of atrix V, we can notice an increase of the null-space tracking errors. In our previous work, 8 we had claied that discontinuous V can produce instability, since the differentiation of discontinuous V results in an unbounded signal. In practice, nuerical differentiation gives bounded signal, and causes only perforance degradation of the null-space tracking. However, the perforance degradation increases at higher joint velocities q. In our experient, joint velocities were rather low; therefore, larger errors can be expected at higher joint velocities. Another advantage of the Givens-rotation-based inial null-space calculation is that it is nuerically less deanding than is the Golub Reunsch algorith. 8. Conclusion This paper considers the stability of the control algoriths for redundant robots using inial null-space force. It was shown that the control algoriths which use the Golub Reunsch based SVD, causes the perforance degradation of the null-space control schee. We proposed a solution based on SVD calculation using Givens rotations. The proposed control schee was tested on the siulation of the -DOF obile anipulator syste. The priary task was the end effector trajectory tracking, while avoiding the obstacles as a secondary subtask. The results show good nuerical stability and shorter coputational cycle 5 4 TASK ERROR 5 2 4 6 8. MNS ERROR. 2 4 6 8 4 2 2 COMPENSATION SIGNAL 4 2 4 6 8 Fig. 9. Experiental results with MNSVC using Golub Reunsch SVD algorith.
52 Coparison of null-space and inial null-space control algoriths copared to the SVD based on Golub Reunsch algorith. Siilar results were obtained with real ipleentation on 7-DOF robot. However, coparison of the results obtained with NSVC and MNSVC algorith shows no significant difference. Additionally, the coputational buen with MNSVC is higher. In ost cases, the secondary tasks are related to obstacle avoidance, singularity avoidance, torque optiization, etc., where good tracking in the null-space is not of priary iportance. More iportant is the closedloop stability of the overall control schee. Velocity-based null-space control in both schees guarantees stability, as long as the null-space feedback gain is sufficiently high. On the contrary, if the torque-based null-space control has to be used, inial null-space-based control algoriths is the only solution that assures stable operation. References. H. Asada and J.-J. E. Slotine, Robot Analysis and Control (Wiley, New York, 986). 2. P. H. Chang, A closed-for solution for inverse kineatics of robot anipulators with redundancy, IEEE J. Robot. Auto. RA-3, 393 43 (987). 3. P. Hsu, J. Hauser and S. Sastry, Dynaics control of redundant anipulators, Proceedings of the IEEE Conference on Robotics and Autoation, Philadelphia, PA (988) pp. 83 87. 4. R. Featherstone and O. Khatib, Load independance of the dynaically consistent inverse of the Jacobian atrix, Int. J. Robot. Res. 6(2), 68 7 (997). 5. O. Khatib, Real-tie obstacle avoidance for anipulators and obile robots, Int. J. Robot. Res. 5, 9 98 (986). 6. O. Khatib, A unified approach for otion and force control of robot anipulators: The operational space forulation, IEEE Trans. Robot. Auto. 3(), 43 53 (987). 7. O. Khatib, The arguented object and reduced effective inertia in robot systes, Proceedings of the IEEE Conference on Robotics and Autoation, Atlanta, GA (988) pp. 24 247. 8. C. W. Kennedy and J. P. Desai, Model-based controller for the Mitsubishi PA- robot ar: Application to robot-assisted surgery, Proceedings of the IEEE Conference on Robotics and Autoation, New Orleans, LA (24). 9. J. Y. S. Luh, Convetional controller design for industrial robots A tutorial, IEEE Trans. Syst., Man, Cybern. SMC- 3(3), 298 36 (983).. J. Y. S. Luh, M. W. Walker and R. P. C. Paul, Resolved acceleration control of echanical anipulators, IEEE Trans. Auto. Control AC-25(3), 486 474 (98).. A. A. Maciejewski and C. A. Klein, The singular value decoposition: Coputation and application to robotics, Int. J. Robot. Res. 8(6), 63 79 (989). 2. A. A. Maciejewski, Kinetic liitation on the use of redundancy in robotic anipulators, IEEE Trans. Robot. Auto. 7(2), 25 2 (99). 3. J.-C. Nash, A one-sided tranforation ethod for the singular value decoposition and algepraic eigenproble, Coput. J. 8(), 74 76 (974). 4. C. Natale, B. Siciliano and L. Vilani, Spatial ipedance control of redundant anipulators, Proceedings of the IEEE Internatinal Syposiu on Robotics and Autoation (ICRA 99), Detroit, MI (999). 5. B. Neec, Force control of redundant robots, In: Preprits of the 5th IFAC Syposiu on Robot Control (SYROCO 97) (M. Guglieli, ed.), Nantes, France (997) pp. 25 29. 6. B. Neec and L. Zlajpah, Null velocity control with dinaically consistent pseudo-inverse, Robotica 8, 53 58 (2). 7. B. Neec and L. Zlajpah, Experients with force control of redundant robots in unstructured environent using inial null-space forulation, J. Adv. Coput. Intell. 5(5), 263 268 (2). 8. B. Neec, L. Zlajpah and D. Orcen, Stability of null-space control algoriths, Proceedings of the 2th RAAD Workshop, Cassino, Italy (23). 9. D. N. Nenchev, Redundancy resolution through local optiization: A review, J. Robot. Syst. 6(6), 769 798 (989). 2. Y. Oh, W. K. Chung, Y. You and I. Suh, Experients on extended ipedance control of redundant anipulator, Proceedings of the IEEE/RJS International Conference on Intelligent Robots and Systes, Victoria, Canada (998) pp. 32 325. 2. D. Orcen, L. Zlajpah and B. Neec, Autonoous otion of a obile anipular using a cobined torque and velocity control, Robotica 22, 623 632 (24). 22. K. O Neil and Y. C. Chen, Instability of pseudoinverse acceleration control of redundant echaniss, Proceedings of the IEEE International Conference on Robotics and Autoation, San Francisco, CA (2) pp. 2575 2582. 23. K. O Neil, Divergence of linear acceleration-based redundancy resolution schees, IEEE Trans. Robot. Auto. 8(4), 625 63 (22). 24. J. Park, W. Chung and Y. You, Weighted decoposition of kineatics and dynaics of kineatically redundant anipulators, Proceedings of the IEEE Conference on Robotics and Autoation, Minneapolis, MN (996) pp. 48 486. 25. J. Park, W. Chung and Y. You, Characterization of instability of dynaic control for kineatically redundant anipulators, Proceedings of the IEEE Conference on Robotics and Autoation, Washington, DC (22) pp. 24 245. 26. D. Sion, K. Kapellos and B. Espiau, Control laws, task and procedures with ORCCAD: Application to the control of an underwater ar, Proceedings of the International Advanced Robotics Prograe Workshop on Underwater Robotics, Toulon, France (996) pp. 27 76. 27. T. Yoshikawa, Foundations of Robotics: Analysis and Control (MIT Press, Cabridge, MA, 99).