An analysis of the operational space control of robots*

Transcription

1 SIMech technical reports (SR_V_N3 MCH) Volume Number 3 Jul-Sep An analysis of the operational space control of robots* N. D. Vuong, H. A. Marcelo,. M. Lim, an S. Y. Lim Abstract heoretically, the operational space control framework [] can be regare to be the most avance control framework for reunant robots. However, in practice, the control performance of this framework is significantly egrae in the presence of moel uncertainties an iscretising effects. Using the singular perturbation theory, this paper shows that the same moel uncertainties can create ifferent effects on the task space an joint space control performance. From the analysis, a multi-rate operational space control was propose to minimise the effects of moel uncertainties on the control performance an while maintaining the avantages of the original operational space framework []. In this paper, we present a stability analysis of the multi-rate operational space control framework using the Lyapunov s irect metho. Keywors: Operational space control, Lyapunov, Digital controller, Moel Uncertainties BACKGROUND Generally, most operational space (or task space) control approaches relate to irectly closing the control loop in the operational space. ask space commans from the operational space controller are then translate into joint space commans through kinematic transformations. Operational space control approaches can be ivie into three groups base on the way they hanle the reunancy [3]. he first group resolves the reunancy at the velocity level [4-5] while the secon [6-7] an the thir groups [] are base on the acceleration. he main ifference between the secon an the thir group is whether the task-space an null-space are kinematically (secon) or ynamically (thir) ecouple. Because of the ynamically ecoupling property, the thir approach sometime has been referre as the force-base operational space control [3]. heoretically, the force-base operational space control is one of the most avance control framework for reunant robots. One main reason is because it uses the system s inertial matri to weight the pseuo-inverse solution. hus, the solution provie by this framework is an optimal solution since the instantaneous kinetic energy is minimise along the path [8]. In other wors, the force-base operational space framework provies a natural choice to ecouple the task-space ynamics from its internal (or null-space) ynamics. In orer to aress the nonlinear effects ue to the link inertia, the gravity an so on in task space, Khatib introuce the concept of task-space ynamics []. He also suggeste a moelbase PD controller to achieve asymptotical performance. It is worth noting that this asymptotical performance is only vali when the robot moel is accurately known an the controller is continuously implemente. However, the assumption of perfect knowlege is always violate in practice, an therefore the control performance can be significantly egrae ue to the imperfect moel as has been eperimentally shown in [-3]. In orer to overcome this uncertainty issue, a number of approaches such as robust control approaches, aaptive control approaches an so on, have been propose in the literature. his paper looks at the moel uncertainty issue from a ifferent aspect: a multi-rate operational space control []. his approach has also been referre to as the inner-outer loop approach in the literature although the initial motivation is ifferent. One of the main initial motivations for the inner-outer loop approach comes from the fact that most of the inustrial robot come with a motion controller at each joint [9]. As a result, the task-space commans can only be realise by creating an outer loop in the operational space. On the other han, the motivation in our case is the inaeuate performance when we implemente the conventional operational space controller [] on our eperimental platform, the Mitsubishi 7-egreeof-freeom (DOF) (from now on, the force-base operational space control introuce by Khaib [] will be referre to as the conventional operational space control). One eplanation for this poor performance is the inaccuracy of the task space inertia matri [3]. It is also worth noting that the subject of moel uncertainties in control has receive a consierable attention from the research community. For eample, Qu [] prove that the continuous PD computetorue is robust with respect to the unknown ynamics. Moreover, the stability of the error system can be mae asymptotically stable with a finite-high control gains if the static balancing torue is known. In practice, however, the control gains have been shown to have upper limits ue to the presence of moel uncertainties an iscretising effects []. Department of Mechanical Engineering, he National University of Singapore, Singapore * his paper has been publishe in IEEE ICRA, Alaska, USA, 3-8 May, pp

2 N. D. Vuong et al So far, it is not known whether these upper limits of the control gains can have any averse effects on the control performance. By analysing the closeloop systems of two control schemes (the igital PD joint space compute-torue control an the igital PD task space compute torue control), we provie a etaile eplanation why the theoretical more avance force-base operational space control cannot perform well in the presence of moel uncertainties. OBJECIVE Base on the above analysis, a multi-rate control structure which was propose in [] is reviewe in the net section. Note that this control structure is not new in the sense that it has been mentione in some previous works such as [9,-3]. he contribution here is that the feeback linearisation concept has been shifte from the operational space [] into joint space in orer to minimise the effects (if possible) of the moel uncertainties. Details of the multi-rate control structure are given in section -3. It is also important to point out that in most of the previous works, the robot moel is either perfectly known [] or ignore [9,3]. As a result, the performance of the inner-loop can be egrae if the nonlinear effects of the robot moel are significant. he main contributions of this paper are as follows: An eplicit eplanation on how the moel uncertainties affect the control performance of the force-base operational space control. An analysis on the stability of the multi-rate operational space control framework uner the presence of moel uncertainties. he multi-rate operational space controller an its stability are verifie eperimentally. he rest of the paper is presente as follows: firstly, the iscrete high-gain compute-torue controls are analyse by the singular perturbation theory. Net, the multi-rate operational space control is reviewe an finally, the stability of the propose controller is given using Lyapunov s irect metho. 3 MEHODOLOGY 3. High-gain Compute-torue Control he following iscussion is motivate from []. Consier a simple rigi ynamic moel of an n-dof robot (without gravity an joint friction) in joint space an task space: A ( )&& + C (&, ) =Γ () Λ ( ) && + μ(, ) = F () where an are the generalise coorinate, Γ an F are the generalise force in joint space an task space accoringly. For simplicity, let us only consier non-reunant robots at singular-free configuration in this section. he relationship between the joint space ynamics an task space ynamics can be state as follows []: Λ= J AJ (3) μ = J C Λ J & & (4) Γ= J F (5) Consier the two set-point PD compute-torue controllers in joint space an task space as follows: ( & ( )) Γ = Au + C = A k + k + C v p ( & ( )) (6) F = Λ u + μ =Λ kv+ kp + μ (7) where an are the esire set point, kv,, k p, are the control gains an ACΛ,, an μ are the estimate/ientifie ynamic moel of the robot. he close-loop euations in joint space an task space become: && = A As A C,&& = Λ Λ s Λ μ where: kp kp kv =, =, kv =, = kv kv s = & s = ( ), & ( ) (8-9) (-) After introucing a fast time scale, τ = t /, the joint space an task space close-loop system become: = & t s A As A C τ = + = & t s =Λ Λ s + Λ μ τ ( & ) ( & ) (-3) At high-gain i.e.,, (-3) reuces to the fast reuce subsystem by the singular perturbation theory [4]: s = A τ As = s an s =Λ Λ τ s =s (4) 6

3 An analysis of the operational space control of robots Note that the inertia matrices A, Λ are always positive efinite, thus if the estimate/ientifie inertia matrices A, Λ is also positive efinite (please refer to [5] for etaile iscussion on how to obtain an positive efinite inertia matri), the eigenvalues of, will be all positive [6]. As a result, s an { } s ten to eponentially as iscusse in []. As is seen from above analysis, as long as the control gains can be increase, the effects of the moel uncertainties Es. (-3): = & D = Λ D A C & μ (5) on the close-loop response can be mae arbitrary small an the behaviour of the close-loop systems Es. (8-9) can be efine by ajusting an. Practically, because the control laws Es. (6-7) are usually implemente using igital computers, thus, the control gains will have upper limits as iscusse in [] an section IV of [7]. his observation raises a uestion on how these gain s limits restrict the response of the close-loop systems Es. (8-9) in practice. 3. Discrete High-gain Compute orue Control Before iscussing the effects of the iscrete high-gain compute-torue control in joint space an task space, let us summarise the uestion in han again:. Assume that we have an ientifie ynamic moel of the robot in joint space ( AC)., he euivalent task space ynamics can be obtaine using Es. (3-4).. Let the task in joint space an task space be eactly the same i.e. = F ( ), where KIN F KIN is the forwar kinematics of the robot. In aition, let us assume that the kinematics moel is accurately known. 3. Let the control laws Es. (6-7) be igitally implemente with the same sampling perio, an assume that the control gains are chosen high enough so that the close-loop systems can be approimate by E. (4). he uestion we are intereste in here is how the responses of the close-loop systems Es. (-3) will be. Let us first consier the following Lemma: Lemma : Uner the above assumptions (,, an 3), the upper limits of the control gains of the joint space an task space controller Es. (6-7) are the same. Proof: Note that E. (4) can be rewritten as follows: s =A As τ t = + & & A Au (6) s =Λ Λs τ t = + & & Λ Λu (7) he iscrete forms of the above euations, uner the assumption that the computation time of the control law is negligible, are: ( e ) e ( e ) ( e ) e + ( e ) + + k [ + ] k [ ] k [ ] = k [ ] & + & Qk [ + ] = Φ Qk [ ] an (8) e ( e ) + + ( e ) [ k + ] [ ] k [ k ] = [ ] k & + ( e ) e ( e & + ) X[ k + ] = Φ X[ k] (9) where is the sampling perio. Substitute Es. (3-5) into leas to: =Λ Λ= JA AJ = JJ () hus, Φ can be rewritten as: e ( e ) + + ( e ) J Φ = J ( e ) e + ( e ) J J J J = Φ J J Note that by similar matri property, Φ () an Φ have the same set of eigenvalues [6]. Because the stability of Es. (8) an (9) can only be guarantee Φ Φ if an only if the eigenvalues of an are insie the unit circle [8] (this is where the upper limits of the control gains occur), E. () implies that the upper limits of the control gains for both systems are the same. From Lemma, it is clear that the responses of the close-loop systems Es. (-3) will now epen on how significant the isturbances E. (5) are. he reason is because, cannot be arbitrarily reuce to zero to eliminate the effects of the moel uncertainties, as in the continuous case. o see the effects of moel uncertainties on the close-loop systems Es. (-3), let us further epan the isturbance terms E. (5): = & D A C ( ) () D ( ) = & Λ μ = J & A C + I JA AJ J && (3) 7

4 N. D. Vuong et al As is seen, the joint space close-loop system E. () is isturbe by E. () an the responses can be transforme to the operational space using the kinematics relationship between the joint space an task space. However, if the control is one in task space, the close-loop system E. (3) has to cope with the isturbance E. (3) which is the result of the jointspace isturbance E. () multiplie by the Jacobian. Moreover, the uncertainties of the inertia matri also appear as an etra term in the isturbance euation E. (3). As a result, if the kinematics moel of the robot happens to magnify the moelling errors, the control performance of the task space controller E. (7) can be much worse than the one in joint space E. (6). he main reason is because the control gains cannot be further increase to compensate for the moel uncertainties. o verify the above observation, let us consier the simulation of a -DOF robot as shown in Fig.. better to use this ynamic moel in joint space rather than task space unless one can guarantee that the isturbance in E. () is much larger than the one in E. (3). (a) he joint space controller E. (6) for some w y m= ( kg) l = ( m).( m) y= lsin( ), J = lcos( ) A= I 7.68, zz = A= 66.7 C=, C = =.( s) = π /6 y = Fig.. One-DOF arm. For simplicity, let us choose the control gains k, k as (Hurwitz polynomial): p v, kv = w= kp = w = (4) he control laws for joint space an task space are Es. (6) an (7) accoringly. Base on the above iscussion, the upper limit of the control gains is w < for both joint space an task space. Figure (a) shows the response of the joint space controller E. (6) for some w (the responses in task space are similar as the previously prove). his simulation was one using SimMechanics oolbo uner Mat- LAB/Simulink environment. Clearly that when w is near to the theoretical unstable value (), chattering occurre. In orer to evaluate the control performances, the ifference between the task space responses ye = y y is plotte in Fig. (b). Here, y = lsin( ) is the response of the controller E. (6) an y = lsin( ) is the response of the controller E. (7). As is seen, the ifference ye tens to be negative which implies that y goes to y faster. In other wors, uner the same control gains (the maimum gains that the iscrete high-gain system can take), the PD joint space controller E. (6) provies a better response in comparison to the task space controller E. (7). In conclusion, the analysis on this section inicates that for an inaccurate ientifie robot moel, it is (b) he ifference between the task space responses Fig.. System responses with various gain. 4 RESULS & DISCUSSION As iscusse in the previous section, if the joint space isturbance E. () is smaller than the task space isturbance E. (3), the compute-torue control shoul be one in the joint space to minimise the effects of moel uncertainties on the control performance. A natural uestion shoul be raise is how to get the joint space commans from tasks which are specifie in the operational space. As point out in [9], the inverse kinematics approach has several isavantages such as it ignores the robot ynamics an the compleity is significantly increase for reunant robots. On the other han, the operational space control framework [] is suppose to take care of the robot ynamics an ynamically ecouple the null space from the task space. o maintain the avantages of the operational space framework an still minimise the moel uncertainties on the control performance, the following multi-rate operational space control has been propose []: Outer loop: he operational space comman force is compute as in [] using the ientifie ynamics moel as a reference []. his task space comman is then applie to the joint space ientifie moel to get the joint acceleration comman: 8

5 An analysis of the operational space control of robots ( & & ) ( ) && = J u J + I JJ A τ null ( ) ( ) (5) u = && + KV & & + KP (6) where KV, KP is the control gains in task space, an: ( ) J = A J JA J (7) It is worth noting that E. (7) is actually a inertia-weighte pseuo-inverse at the acceleration level [8]. hus, E. (7) will give a joint space response with respect to a task space comman {&&, &, } through the ientifie ynamic moel Â. In other wors, the purpose of the outer loop is to transform the task space comman to the joint space comman using the ientifie ynamic moel. he output of this outer loop is then realise by the PI compute-torue control at the inner loop as below. Inner loop: a PI compute torue controller is use to control the joint velocity of the robots. he use of the ynamics moel here will enhance the performance as in [5]. he input of this controller is the reference joint velocities. he controller can be state as: Γ= Au + C + g +Γ Fric u = && + K( & & ) + K ( & & ) t I (8) (9) where K, K I are the control gains in joint space. he esire joint velocity & can be obtaine by integrating the esire joint acceleration E. (5): t+δt t &( +Δt) t &&( ) t t (3) with the initial conition as the current { &,}. Clearly, if the inner velocity control loop is able to bring the manipulator from the current state { t &( ), t ( )} to { t &( +Δ t), t ( +Δt)} after Δ t (sec), the behaviour of the robot will be eactly etermine by the ientifie ynamic moel as epicte in Fig. 3. Because the assumption that the inner velocity control loop can change the system states in Δ t (sec) is usually violate in practice, an outer loop, which is the force-base operational space control, is always necessary to ensure the task space performance. he efficiency of the propose multi-rate operational space control has been eperimentally verifie on the Mitsubishi 7-DOF PA manipulator as escribe in the following section. 4. Eperiment est-be he propose controller has been implemente on the Mitsubishi 7-DOF PA manipulator. In orer to achieve real-time torue control capability (which is necessary for the inner-loop controller), the original controller of the PA has been replace by our custom controllers. QNX Neutrino Real-time Operate System (6.3) has been use to implement the above multi-rate control laws as epicte in Fig. 3. Note that because the robot is reunant, the following simple null-space controller is use throughout this paper: τ = K & V null nd V = ( rest ) KnP ( rest ) where rest { X&&, X&, X, F} Operational space control KHz X is some preferre joint configuration. esire F C F J, sensor (3) (3) Fig. 3. he propose controller: (ot frame): outer loop, (ot-ash frame): inner loop. 4. Eperiment Results τ C τ O Compute-torue Forwar τ ynamics Robot & esire Null-Sp ace control Forwar kinematics 5KHz control In orer to evaluate the control performance, the en-effector of the manipulator was commane to move. m in the y-irection of the base frame in s from the same initial configuration (Fig. 4(a)) using two ifferent controllers: (I): the conventional operational space control introuce by Khatib [] an (II: the propose multi-rate operational space control. Both controllers has been tune the first sign of instability appears i.e. chattering occurs at any control variables. Fig. 4. (a) Eperiment setup, (b) tracking error in X-irection, (c) tracking error in y-irection, an () tracking error in z-irection. 9

6 N. D. Vuong et al As is seen in Fig. 4, the tracking performance of the conventional operational space control (I) is poorer than the one using the propose controller (II). One eplanation as iscusse in section is because the moelling errors, such as inertial parameters an joint frictions, are magnifie through the Jacobian as in E. (3). 4.3 Stability Analysis of the Multi-rate OS Control he iscussion from section an the eperimental results from the previous section is consistent in justifying the usefulness of the propose multi-rate operational space controller over the conventional one. However, it is still necessary to investigate the stability of the propose controller, at least for the continuous case. It is important to stress that the stability analysis in this section only serves as a necessary conition for the usefulness of the propose controller because we o not account for iscretising effects, signal noise an so on. If the control law is igitally implemente, the performance of the close-loop system will now epen on how high the control gains can take as iscusse in section. Note that only a sketch of proof is given here ue to the limite space Problem Statements Consier the rigi ynamic moel of an n-dof robot as follows: A ( )&& + C (&, ) + g ( ) +Γ Fric + D Γ =Γ (33) where { A, C, g, D τ } are inertial matri, Coriolis-Centrifugal, gravity, friction torue an unknown isturbance in joint space. By applying the control law Es. (5-9), the close-loop control becomes: && + KV& + KP = Jw& w & + A AKw + A AK Iz = A A && + A H z& = w (34) I & K P K V JA AK JA AK I & = t w AAKACAAAK AAK w I AC z I z JA A JA H && + + ( A C+ A A+ A K) w + ( A C) z+ A A && + A H X& = X + B where: (35) (36) && = && &&, & = & &, = w = & = & &, w& = && = && && H = C + g +Γ + D C g Γ Fric Γ Fric (37) Here, we are intereste in the stability property of the euilibrium & w z = [ ] of e the nonlinear system E. (34). Note that there are etra terms ( A C A A A K ) in the matri, an they have been cancelle out later in vector B. he purpose of these terms is to simplify the analysis for the nominal system as shown in the net section. o analyse the stability of E. (34), we aopt the methoology propose by Khalil [] that is: Firstly, the asymptotic stability property of the nominal system X & = X is stuie. Net, the solution of the overall system E. (34) is shown to be uniformly ultimately boune Stability of the Nominal System Consier the following Lyapunov function caniate inspire by []: V = X PX (38) where: K v I I I P = A A A K + A (39) is positive efinite when kv >, k >, >. he erivative of V is (after making use of the skew-symmetric property of the inertia matri, X ( A& C C ) X = ): K P JA AK JA AK I & K P KV I JA AK JA AKI & V& = =X QX w AK K AK + w I z AK AK z I (4) After some manipulation, it can be shown that Q > when: kp >, kv > ( / 4) kp +, > > ( / )( kp + kv kp + ( kp kv + ) ) > ( ) k kh / ( ( ) λ ( ( ) λ) 4 λ ) (4) where λ = λ min ( A) is the smallest eigenvalues of Â. Assume that the inuce norm of the Jacobian is boune by J < kj then k H = kjλma ( A ) λma ( A). As is seen, if the control gain k is fie (i.e. after the inner-loop control is tune), E. (4) can always be satisfie by increasing the task space gains { kp, kv }. As a result, the nominal system X& =X is eponentially stable because: & V = X QX < λmin ( Q) X, λmin ( Q) > (4)

7 An analysis of the operational space control of robots Stability of the Overall System Because the isturbance B is a function of joint position an acceleration, using the similar approach as in [], it can be shown that: B < ζ + ζ X + ζ X 3 (43) where ζ, ζ, ζ 3 > are the system parameters. Using the same Lyapunov function E. (38), V & now becomes: & V= X QX+ X B X ( ζ+ ( ζ λmin( Q)) X + ζ3 X ) (44) hus, by applying Lemma 3.5 in [], the overall system is uniformly ultimately boune. Note that the purpose of this stability analysis is only to show that the propose control law can be stabilise by a proper choice of control gains. However, in practice, the size of the uniform/uniform ultimate boun cannot be mae arbitrarily small because of the upper limits on the control gains. his propose controller performe better than the conventional operational space control because the moel uncertainties has been shifte from task space to joint space as the iscussion in section -3. Intensive eperiments ha been one in orer to valiate the performance of the propose controller []. 5 CONCLUSION In this paper, we ha eplicitly shown that the control performance of the conventional operational space control can be significant egrae ue to moel uncertainties an iscretising effects. As a result, the compute torue techniue shoul be one in joint space to avoi magnifying the moelling errors through the robot kinematics. In orer to maintain the avantages of the force-base operational space control, a multi-rate operational space controller was propose. Eperimental results showe significant improvements in comparison to the conventional one. Stability analysis ha been carrie out to show that the propose controller is stable in the continuous omain. Since the joint space control of the propose controller is a simple PI controller, further improvements on the inner loop control can be one in orer to enhance the overall performance of the system. 6 INDUSRIAL SIGNIFICANCE he control performance of the force-base operational space controller uner the presence of moel uncertainties an igitise effects was analyse. he focus of this work was only on the force-base operational space control because this control framework can be consiere to as the more avance control framework for reunant robots. Eperimental results in Section 4 inicate that the accuracy of the robot ynamic moel plays an important role on the operational space control performance. Since one oes not have access to the eact ynamic moel, the ifferences between the estimate an real moel can significantly egrae the operational space control performance. Further investigation on the igitise effects of the igital controllers inicate that the maimum control gains for both joint-space compute-torue an task-space compute-torue control are the same. his fining is of crucial importance because it has provie a conclusive evience for why the theoretically more avance force-base operational space controllers ehibit a significant epenence on the accuracy of the ynamic moel. By making use of the singular perturbation theory, eplicit conitions of when the moel uncertainties most affect the operational space controller the most were also erive. REFERENCES [] O. Khatib, A unifie approach for motion an force control of robot manipulators: he Operational Space Formulation, IEEE J. Robot. Autom., vol. RA-3(): pp , 987. [] V. Ngoc Dung, et al., Multi-rate perational space control of compliant motion in robotic manipulators, IEEE SMC, San Antonio, eas, US, 9. [3] J. Nakanishi, Operational space control: A theoretical an empirical comparison, Int. J. 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