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Experimental Control of Flexible Robot Manipulators 167 Figure 8. a) Contribution to the torque control τ 1 (rigid) with the LQR combined control squeme. b) Contribution to the torque control τ 1 (rigid) in the sliding-mode case 4. The IDA-PBC method 4.1 Outline of the method The IDA-PBC (interconnection and damping assignment passivity-based control) method is an energy-based approach to control design (see (Ortega & Spong, 2000) and (Ortega et al., 2002) for complete details). The method is specially well suited for mechatronic applications, among others. In the case of a flexible manipulator the goal is to control the position of an under-actuated mechanical system with total energy: 1 1 (, ) () () 2 T Hqp pM qp Uq − =+ (16) where n q ∈ , n p ∈ , are the generalized positions and momenta respectively, M(q)=M T (q)>0 is the inertia matrix and U(q) is the potential energy. Robot Manipulators 168 If it is assumed that the system has no natural damping, then the equations of motion of such system can be written as: 0 0 0 () q n p n H I q u H I pGq ∇ ⎛⎞ ⎛⎞ ⎛⎞ ⎛ ⎞ =+ ⎜⎟ ⎜⎟ ⎜⎟ ⎜ ⎟ ∇ − ⎝⎠ ⎝ ⎠ ⎝⎠ ⎝⎠ (17) The IDA-PBC method follows two basic steps: 1. Energy shaping, where the total energy function of the system is modified so that a predefined energy value is assigned to the desired equilibrium. 2. Damping injection, to achieve asymptotic stability. The following form for the desired (closed-loop) energy function is proposed: 1 1 (, ) () () 2 T ddd Hqp pM qpUq − =+ (18) where 0 T dd MM=> is the closed-loop inertia matrix and U d the potential energy function. It will be required that U d have an isolated minimum at the equilibrium point q * , that is: * arg min ( ) d qUq= (19) In PBC, a composite control law is defined: (, ) (, ) es di uuqp uqp=+ (20) where the first term is designed to achieve the energy shaping and the second one injects the damping. The desired (closed-loop) dynamics can be expressed in the following form: () (, ) (, ) qd dd pd H q Jqp Rqp H p ∇ ⎛⎞ ⎛⎞ =− ⎜⎟ ⎜⎟ ∇ ⎝⎠ ⎝⎠ (21) where 1 1 2 0 (, ) T d dd d MM JJ M MJqp − − ⎛⎞ =− = ⎜⎟ − ⎝⎠ (22) 00 0 0 T dd T v RR GK G ⎛⎞ == ≥ ⎜⎟ ⎝⎠ (23) represent the desired interconnection and damping structures. J 2 is a skew-symmetric matrix, and can be used as free parameter in order to achieve the kinetic energy shaping (see Ortega & Spong, 2000)). The second term in (20) , the damping injection, can be expressed as: T di v p d uKGH=− ∇ (24) where 0 T vv KK=>. Experimental Control of Flexible Robot Manipulators 169 To obtain the energy shaping term u es of the controller, (20) and (24), the composite control law, are replaced in the system dynamic equation (17) and this is equated to the desired closed-loop dynamics, (21): 1 1 2 0 0 0 0 (, ) qqd n d es p pd n d HH I MM u HH I G MM J qp − − ∇∇ ⎛⎞⎛⎞ ⎛ ⎞ ⎛⎞ ⎛⎞ += ⎜⎟⎜⎟ ⎜ ⎟ ⎜⎟ ⎜⎟ ∇∇ − − ⎝⎠ ⎝⎠ ⎝⎠ ⎝ ⎠⎝⎠ 11 2 es q d q d d Gu H M M H J M p −− =∇ − ∇ + (25) In the under-actuated case, G is not invertible, but only full column rank. Thus, multiplying (25) by the left annihilator of G, G ┴ G=0 , it is obtained: { } 11 2 0 qdqd d GHMMHJMp ⊥−− ∇− ∇ + = (26) If a solution (M d ,U d ) exists for this differential equation, then u es will be expressed as: 111 2 ()( ) TT es q d q d d uGGGHMMHJMp −−− =∇−∇+ (27) The PDE (26) can be separated in terms corresponding to the kinetic and the potential energies: { } 111 1 2 () ()2 0 TT qdqdd G pMp MM pMp JMp ⊥− −− − ∇−∇+= (28) { } 1 0 qdqd GUMMU ⊥− ∇− ∇ = (29) So the main difficulty of the method is in solving the nonlinear PDE corresponding to the kinetic energy (28). Once the closed-loop inertia matrix, M d , is known, then it is easier to obtain U d of the linear PDE (29), corresponding to the potential energy. 4.2 Application to a laboratory arm The object of the study is a flexible arm with one degree of freedom that accomplishes the conditions of Euler-Bernoulli (Fig.9). In this case, the elastic deformation of the arm w(x,t) can be represented by means of an overlapping of the spatial and temporary parts, see equation (1). Figure 9. Photograph of the experimental flexible manipulator Robot Manipulators 170 If a finite number of modes m is considered, Lagrange equations lead to a dynamical system defined by m+1 second order differential equations: 00 00 () () 01 0 () () 0(0) t f ii qt qt I u K qt qt I ϕ ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ += ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ′ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ (30) where q 0 (a 1x1 vector) and q i (an mx1 vector) are the dynamic variables; q 0 (t) is the rigid generalized coordinate, q i (t) is the vector of flexible modes, I t is the total inertia, K f is the stiffness mxm matrix that depends on the elasticity of the arm, and it is defined as K f =diag( 2 i ω ), where ω i is the resonance frequency of each mode; φ’ are the first spatial derivatives of φ i (x) evaluated at the base of the robot. Finally u includes the applied control torques. Defining the incremental variables as d qqq=− , where q d is the desired trajectory for the robot such that 0 0 d q ≠ , 0 0 d q = and q id =0, then the dynamical model is given by: 00 0 00 () 01 () 0 () 0(0) () d t f i i qt q I qt u K qt I qt ϕ + ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ ⎛⎞ += ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ ′ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ ⎝⎠ (31) The total energy of the mechanical system is obtained as the sum of the kinetic and potential energy: 1 11 (, ) 22 TT Hqp pM p qKq − =+ (32) where 0 () T ti pIqIq= and M and K are (m+1)x(m+1) matrices 00 0 0 0 t f I MK K I ⎛⎞ ⎛⎞ == ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ The point of interest is * (0,0)q = , which corresponds to zero tracking error in the rigid variable and null deflections. The controller design will be made in two steps; first, we will obtain a feedback of the state that produces energy shaping in closed-loop to stabilize the position globally, then it will be injected the necessary damping to achieve the asymptotic stability by means of the negative feedback of the passive output. The inertia matrix, M, that characterizes to the system, is independent of q , hence it follows that it can be chosen J 2 =0, (see (Ortega & Spong, 2000)). Then, from (28) it is deduced that the matrix M d should also be constant. The resulting representation capturing the first flexible mode is 12 23 d aa M aa ⎛⎞ = ⎜⎟ ⎝⎠ where the condition of positive definiteness of the inertia matrix, lead to the following inequations: 2 1132 0aaaa>> (33) Experimental Control of Flexible Robot Manipulators 171 Now, considering only the first flexible mode, equation (29) corresponding to the potential energy can be written as: 2 21 32 11 01 (0) ((0)) dd t aaU U aa q Iq q ϕ ϕ ω ⎛⎞ ′ −∂ ∂ ′ +− = ⎜⎟ ∂∂ ⎝⎠ (34) where 000d qqq=− is the rigid incremental variable and 11 0qq=− is the flexible incremental variable. The equation (34) is a trivial linear PDE whose general solution is 22 2 2 13 2 1 001 2 21 21 ( (0) ) () 2( (0) ) ( (0) ) tt d Ia a I UqqqFz aa aa ωϕ ω ϕϕ ′ − =− − + ′′ −+ −+ (35) 01 zqq γ =+ (36) 32 21 ((0)) ((0)) t Ia a aa ϕ γ ϕ ′ − = ′ −+ (37) where F is an arbitrary differentiable function that we should choose to satisfy condition (19) in the points q * . Some simple calculations show that the necessary restriction (0) 0 qd U∇= is satisfied if ((0)) 0Fz∇=, while condition 2 (0) 0 qd U∇> is satisfied if: 2 1 32 ((0)) F aa ω ϕ ′′ > ′ − (38) Under these restrictions it can be proposed 2 1 () (1/2)Fz Kz= , with K 1 >0 , which produces the condition: 32 (0)aa ϕ ′ > (39) So now, the term corresponding to the energy shaping in the control input (27) is given as 11 10 21 ()( ) TT es q d q d p p uGGGUMMUKqKq −− =∇−∇=+ (40) where 2 2 13 2 11321 2 21 22 11 1 13 2 2 21 () (( (0)) ) ((0)) () ((0)) t p p Iaa a KKaa aa aKaaa K aa ϕ ω ϕ ω ϕ − ′ =−++ ′ −+ −− = ′ −+ The controller design is completed with a second term, corresponding to the damping injection. This is achieved via negative feedback of the passive output T p d GH∇ , (24). As Robot Manipulators 172 1 (1 / 2) ( ) T ddd HpMpUq − =+ , and U d only depends on the q variable, u di only depends on the appropriate election of M d : 10 21di v v uKqKq=+ (41) with 32 1 2 13 2 21 2 2 13 2 ((0)) () ((0)) () t vv vv Ia a KK aa a aa KK aa a ϕ ϕ ′ −+ = − ′ − = − A simple analysis on the constants K p1 and K v1 with the conditions previously imposed, implies that both should be negative to assure the stability of the system. To analyze the stability of the closed-loop system we consider the energy-based Lyapunov function candidate (Kelly & Campa, 2005), (Sanz & Etxebarria, 2007) 22 2 2 2 2 1 03 012 11 1 3 2 0 22 13 2 2 1 2 2 101 101 21 1121((0)) (,) 22 2((0)) 1 () (0) 2 T tt t dd t Iqa Iqqa qa I a a q Vqq pM p U aa a a a Iqq Kqq aa ωϕ ϕ ω γ ϕ − ′ −+ − =+= − ′ −−+ −++ ′ −+ (42) which is globally positive definite, i.e: V(0,0)=(0,0) and (,) 0Vqq> for every (,) (0,0)qq ≠ . The time derivative of (42) along the trajectories of the closed-loop system can be written as 2 03 2 2 11 22 13 2 ( ( (0))( (0))) (,) 0 () t v Iq a a a a q Vqq K aa a ϕϕ ′′ −+ + − =− ≤ − (43) where K v >0, so V is negative semidefinite, and (,) (0,0)qq = is stable (not necessarily asymptotically stable). By using LaSalle's invariant set theory, it can be defined the set R as the set of points for which 0V = {} 0 1 41 01 0 1 0 1 :(,) 0 , & 0 q q RVqqqqqq q q γ ⎧⎫ ⎛⎞ ⎪⎪ ⎜⎟ ⎪⎪ ⎜⎟ =∈ ==∈ += ⎨⎬ ⎜⎟ ⎪⎪ ⎜⎟ ⎜⎟ ⎪⎪ ⎝⎠ ⎩⎭ (44) Following LaSalle's principle, given R and defined N as the largest invariant set of R, then all the solutions of the closed loop system asymptotically converge to N when t →∞. Any trajectory in R should verify: 01 0qq γ += (45) Experimental Control of Flexible Robot Manipulators 173 and therefore it also follows that: 01 0qq γ += (46) 01 qqk γ += (47) Considering the closed-loop system and the conditions described by (46) and (47), the following expression is obtained: 2 1 0 21 ((0))( (0))0 ((0)) f t Kk q Iaa γγω ϕϕ ϕ ′′ +−+ + = ′ −+ (48) As a result of the previous equation, it can be concluded that 0 ()qt should be constant, in consequence 0 () 0qt= , and replacing it in (45), it also follows that 1 () 0qt= . Therefore 01 0qq== and replacing these in the closed-loop system this leads to the following solution: 0 1 () 0 () 0 qt qt ⎛⎞ ⎛⎞ = ⎜⎟ ⎜⎟ ⎝⎠ ⎝⎠ In other words, the largest invariant set N is just the origin (,) (0,0)qq = , so we can conclude that any trajectory converge to the origin when t →∞, so the equilibrium is in fact asymptotically stable. To illustrate the performance of the proposed IDA-PBC controller, in this section we present some simulations. We use the model of a flexible robotic arm presented in (Canudas et al., 1996) and the values of Table 2 which correspond to the real arm displayed in Fig. 9. The results are shown in Figs. 10 to 12. In these examples the values a 1 =1, a 2 =0.01 and a 3 =50 have been used to complete the conditions (33) and (39). In Fig. 10 the parameters are K 1 =10 and K v =1000. In Figs. 11 and 12 the effect of modifying the damping constant K v is demonstrated. With a smaller value of K v , K v =10, the rigid variable follows the desired trajectory reasonably well. For K v =1000, the tip position exhibits better tracking of the desired trajectory, as it can be seen comparing Fig. 10(a) and Fig. 11(a). Even more important, it should be noted that the oscillations of elastic modes are now attenuated quickly (compare Fig. 10(c) and Fig. 11(b)). But If we continue increasing the value of K v, K v =100000, the oscillations are attenuated even more quickly (compare Fig. 10(c) and Fig. 12(b)), but the tip position exhibits worse tracking of the desired trajectory. Property Value Motor inertia, I h 0.002 kgm 2 Link length, L 0.45 m Link height, h 0.02 m Link thickness, d 0.0008 m Link mass, M b 0.06 kg Linear density, ρ 0.1333 kg/m Flexural rigidity, EI 0.1621 Nm 2 Table 2. Flexible link parameters Robot Manipulators 174 Figure 10. Simulation results for IDA-PBC control with K v =1000: (a) Time evolution of the rigid variable q 0 and reference q d __; (b) Rigid variable tracking error; (c) Time evolution of the flexible deflections; (d) Composite control signal The effectiveness of the proposed control schemes has been tested by means of real time experiments on a laboratory single flexible link. This manipulator arm, fabricated by Quanser Consulting Inc. (Ontario, Canada), is a spring steel bar that moves in the horizontal plane due to the action of a DC motor. A potentiometer measures the angular position of the system, and the arm deflections are measured by means of a strain gauge mounted near its base (see Fig. 9). These sensors provide respectively the values of q 0 and q 1 (and thus 0 q and 1 q are also known, since q 0d and q 1d are predetermined). The experimental results are shown on Figs. 13, 14 and 15. In Fig. 13 the control results using a conventional PD rigid control design are displayed: 00 00 ()() P dD d uKq q Kq q=−+− where K P =-14 and K D =-0.028. These gains have been carefully chosen, tuning the controller by the usual trial-an-error method. The rigid variable tracks the reference (with a certain error), but the naturally excited flexible deflections are not well damped (Fig. 13(b)). Experimental Control of Flexible Robot Manipulators 175 Figure 11. Simulation results for IDA-PBC control with K v =10: (a) Time evolution of the rigid variable q 0 and reference q d __; (b) Time evolution of the flexible deflections Figure 12. Simulation results for IDA-PBC control with K v =100000: (a) Time evolution of the rigid variable q 0 and reference q d __; (b) Time evolution of the flexible deflections In Fig. 14 the results using the IDA-PBC design philosophy are displayed. The values a 1 =1, a 2 =0.01, a 3 =50, K 1 =10, K v =1000 and (0) 1.11 ϕ ′ = have been used. As seen in the graphics, the rigid variable follows the desired trajectory, and moreover the flexible modes are now conveniently damped, (compare Fig. 13(b) and Fig. 14(c)). It is shown that vibrations are effectively attenuated in the intervals when q d reaches its upper and lower values which go from 1 to 2 seconds, 3 to 4 s., 5 to 6 s., etc. The PD controller might be augmented with a feedback term for link curvature: 00 00 ()()w PdDdC uKq q Kq q K ′′ =−+−+ where w ′′ represents the link curvature at the base of the link. This is a much simpler version of the IDA-PBC and doesn't require time derivatives of the strain gauge signals. [...]... Kubus, D and Wahl, F.M (20 06) 6D Force and Acceleration Sensor Fusion for Compliant Manipulation Control Int Conf Intelligent Robots and Systems (IROS 20 06) , pages 262 6- 263 1, Beijing, China (Kröguer et al., 20 06) Kröger, T , Kubus, D and Wahl, F.M (2007) Force and acceleration sensor fusion for compliant manipulation control in 6 degrees of freedom Advanced Robotics, 21: 160 3 161 6(14), October (Kröguer... Automatica, 38, pp 585 5 96, 2002 A Sanz and V Etxebarria, Experimental Control of a Two-Dof Flexible Robot Manipulator by Optimal and Sliding Methods Journal of Intelligent and Robotic Systems, 46: 95110, 20 06 A Sanz and V Etxebarria, Experimental control of a single-link flexible robot arm using energy shaping International Journal of Systems Science, 38(1): 61 -71, 2007 180 Robot Manipulators M W Vandegrift,... 1993) Johansson, R and Robertsson (2003), A Robotic Force Control using Observer-based Strict Positive Real Impedance Control IEEE Proc Int Conf Robotics and Automation, pages 368 6- 369 1, Taipei, Taiwan (Johansson and Robertsson, 2003) Khatib, O (1987) A unified approach for motion and force control of robot manipulators: The operational space formulation IEEE J of Robotics and Automation, 3(1):43-53 (Khatib,... the proposed composite control schemes in practical flexible robot control tasks 6 References O Barambones and V Etxebarria, Robust adaptive control for robot manipulators with unmodeled dynamics Cybernetics and Systems, 31(1), 67 - 86, 2000 C Canudas de Wit, B Siciliano and G Bastin, Theory of Robot Control Europe: The Zodiac, Springer, 19 96 R Kelly and R Campa, Control basado en IDA-PBC del péndulo... in free space (from t =5s to t = 6. 2s), a contact 192 Robot Manipulators transition (from t =6. 2s to t = 6. 4s) and a movement in constrained space (from t =6. 4s to t = 9s) The force controller was an impedance one In this case, it can be compared how the observer eliminates the inertial effects and the noise introduced by the sensors Regarding the noise spectra, in Fig 6, the power spectrum density for... Fusion of Force, Acceleration and Position for Compliant Robot Motion Control Phd Thesis, Jaen University, Spain (Gamez, 2006a) Gamez, J., Robertsson, A., Gomez, J and Johansson, R (20 06) Generalized Contact Force Estimator for a Robot Manipulator IEEE Int Conf on Robotics and Automation (ICRA20 06) , pages 4019-4025, Orlando, Florida (Gamez et al., 2006b) Gamez, J., Robertsson, A., Gomez, J and Johansson,... 6D contact force/torque estimator for robotic manipulators with filtering properties was recently proposed in (Gamez et al., 2008b) This work describes how the force control performance in robotic manipulators can be increased using sensor fusion techniques In particular, a new sensor fusion approach applied to the problem of the contact force estimation in robot manipulators is proposed to improve the... Stäubli Platform An RX60 robot with an open control architecture system is used An ATI wrist force sensor is attached to the robot tip The accelerometer is placed between the robot TCP and the force sensor Regarding the second robotic platform (Fig 4), it consisted of a Stäubli manipulator situated in the Robotics and Automation Lab at Jaen University, Spain It is based on a RX60 which has been adapted... and Moments in Multiple Cooperative Robots Trans ASME J of Dynamic Systems, Measurement and Control, 1 26( 2):2 76- 283 (Kumar and Garg, 2004) Lin, S.T (1997) Force Sensing Using Kalman Filtering Techniques for Robot Compliant Motion Control J of Intelligent and Robotics Systems, 135:1- 16 (Lin, 1997) Murray, R M., Li, Z and Sastry S.S., A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca... Gamez, J., Robertsson, A., Gomez, J and Johansson, R (2008) Self-Calibrated Robotic Manipulator Force Observer (In press) Robotics and Computer Integrated Manufacturing (Gamez et al., 2008a) 200 Robot Manipulators Gamez, J., Robertsson, A., Gomez, J and Johansson, R (2008) Sensor Fusion for Compliant Robot Motion Control IEEE Trans on Robotics, 24(2):430-441 (Gamez et al., 2008b) Grewal, M S and Andrews, . flexible robot control tasks 6. References O. Barambones and V. Etxebarria, Robust adaptive control for robot manipulators with unmodeled dynamics. Cybernetics and Systems, 31(1), 67 - 86, 2000 95- 110, 20 06. A. Sanz and V. Etxebarria, Experimental control of a single-link flexible robot arm using energy shaping. International Journal of Systems Science, 38(1): 61 -71, 2007. Robot Manipulators. performance in robotic manipulators can be increased using sensor fusion techniques. In particular, a new sensor fusion approach applied to the problem of the contact force estimation in robot manipulators