Control of industrial robots. Centralized control

Control of industrial robots Centralized control Prof. Paolo Rocco (paolo.rocco@polimi.it) Politecnico di Milano ipartimento di Elettronica, Informazione e Bioingegneria

Introduction Centralized control means that the controller, when determining the input (torue/force) to each joint, takes into account measurements and/or models related to the other joints of the manipulator In general a centralized controller reuires knowledge, at least a partial one, of the mathematical model of the manipulator In the absence of the decoupling effect given by the high reduction ratios (for instance in case of direct drive motors), the use of a centralized control strategy might turn out to be the only viable solution In the following we will discuss a few strategies for centralized control, both under the assumption of perfect knowledge of the model and when some uncertainty has to be taken into account

Introduction Clear improvements with respect to standard non model-based controllers can be achieved:

axonomy Joint space Open loop Computed torue control Closed loop o Regulation problem P gravity compensation o rajectory tracking problem Nominal conditions Inverse dynamic control With uncertainty Robust control & Adaptive control Operational space Closed loop o Regulation problem P gravity compensation o rajectory tracking problem Nominal conditions Inverse dynamic control

Computed torue control In this scheme, the decentralized controller is complemented by a controller that operates in open loop, computing the disturbance torues based on the mathematical model, fed by the position reference signal and its derivatives... md. md md CENRALIZE FEEFORWAR ACION ECENRALIZE CONROLLER d d d MANIPULAOR ECOUPLE m LINEAR SYSEM d d 1 ( ) N 1 1 1 N C(, ) N N g( ) 1 N B d md d d md d Compensation of nonlinearities can be only partial (for instance just the gravitational terms and the diagonal terms of the inertia matrix) Compensation terms can be computed offline, in case of trajectories repeated several times

P plus gravity compensation Let us consider a manipulator whose euations are: B ( ) C(, ) g( ) τ Assume that a constant euilibrium posture d is assigned (regulation problem). We want to find a control law which guarantees the global asymptotic stability of the euilibrium point using Lyapunov method. Let the state be defined as: [ ], d error As a candidate Lyapunov function we take: ( 1 1, ) B( ) K V P If K P is symmetric positive definite, function V is positive definite as well.

P plus gravity compensation Let s take the derivative with respect to time of function V, taking into account that d is constant and using the dynamic model: 1 V B B K 1 ( ) ( ) ( B ( ) C (, )) τ g( ) P ( K ) he first term is zero due to a property of the dynamic model. of the manipulator which we have already proven (skew symmetry of matrix B C). Consider now the following choice for the control law: τ g ( ) K K P with K symmetric and positive definite. We obtain: V K 0 P hen the derivative of V is negative semidefinite.

P plus gravity compensation In order to draw a conclusion about the stability. of the euilibrium state, we need to study the trajectories of the system when V 0. In closed loop it is: B ( ) C(, ) g( ) g( ) K K his is enough to prove that the euilibrium state characterized by: 0, 0 P.... Notice that when V0 it results 0, 0, and then the only trajectory which complies with the system s euations is characterized by: K P or: 0 0 (robot at rest with zero position error) is globally asymptotically stable. La Salle theorem

P plus gravity compensation he control law obtained with the Lyapunov method can be interpreted as a P control in joint space plus a term that compensates for gravitational effects. K d K P τ MANIPULAOR. K P and K diagonal d constant g(.) he theoretical result is of course valid as long as perfect gravity compensation is achieved For a gravity free (or gravity compensated) manipulator a joint space P control yields an asymptotically stable euilibrium state: the proportional gains can be seen as virtual springs while the derivative gains as virtual dampers

P plus gravity compensation he P plus gravity compensation allows to demonstrate an asymptotic result. We do not have control on the time history with which the error goes to zero: i ( 0) i t If we need to have a better control on the time evolution of the error, or if we want to track a time-varying reference position, we need to turn to more complicated schemes.

Inverse dynamics control Consider again the dynamic model of the manipulator: B ( ) n(, ) τ where centrifugal, Coriolis and gravitational terms have been conveniently gathered in vector n: (, ) C(, ) g( ) n Assuming perfect knowledge of the dynamic model, we can take into consideration the control law (inverse dynamics): τ B ( ) y n(, ) Since matrix B is full rank for every robot configuration, the application of the above control law yields: y the system dynamics has then been made linear, completely decoupled and composed of n double integrators (n unitary masses!) y 1 1/s y n... 1/s 1 n

Inverse dynamics control he decoupled system: y can be easily controlled through a decoupled P controller, to which a feedforward acceleration term can be added: y K K P d d is time varying he following closed loop euation is obtained: K K P 0 hen the error is governed by a second order dynamics that can be arbitrarily assigned, on each joint, by suitably selecting the gains in the diagonal matrices K P and K. he inverse dynamics control is a special case of a more general methodology for controlling nonlinear systems called feedback linearization.

Inverse dynamics control With the inverse dynamics control we have full control on the time history with which the error goes to zero: i ( 0) i Assuming matrices K P and K diagonal, the time evolution of the error is governed by eigenvalues that are roots of the polynomial: s Kis KPi 0 t

Inverse dynamics control Inverse dynamics control can be represented through the following block diagram:.. d. d d K K P y B() τ MANIPULAOR.. n(,) compensation terms must be computed online, with small sampling times the recursive NE method is used: τ ΝΕ(,, y ) ( 1 ms) the method reuires perfect cancellation of the terms of the dynamic model, which might be difficult in practice

Robust control In a realistic scenario we have to assume that the compensation of the dynamic model is not perfect. he inverse dynamics control can then be expressed as: ( ) y nˆ (, ) τ Bˆ which yields: B ( ) n(, ) Bˆ ( ) y nˆ (, ) Let us express the uncertainty of the model in the following terms: B Bˆ B, n nˆ n Since matrix B is invertible, we obtain: y with: 1 ( B Bˆ I ) y B 1 n y η 1 ( I B Bˆ η ) y B 1 n

Robust control Adopting the same control law for y as in the ideal case: y K K P d we obtain that the error dynamics is governed by the euation: K K P η which means that the system is still nonlinear and coupled. We need then to add to the P linear control an additional nonlinear term, function of the error, conceived to yield robustness to the project. his term will be designed based on Lyapunov theory.

Robust control From the euation: y η the expression for the second derivative of the error is obtained: y d η efining the system state as: ξ [ ] we obtain the matrix first order differential euation : ( η) ξ Hξ y d H 0 0 I R 0 0 R I n n n n, the system is nonlinear and timevarying

Robust control In the following we will use the concept of norm of a vector and of a matrix. Given a vector x we define the Euclidean norm as: x x x Consider now a matrix A, in general a rectangular matrix. We define norm of A, induced by the definition of the norm of the vector, the uantity: A sup Ax x max x 0 x 1 Ax It can be proven that the norm of the matrix coincides with the maximum singular value of the matrix itself, i.e.: A λ max ( A A)

Robust control We characterize the uncertainty as follows: a) b) c) sup t 0 I B n d 1 Φ < Q M ( ) Bˆ ( ) < α < 1 ( ξ ), d Assumption a) is related to the fact that trajectory planning guarantees a limited acceleration. Assumption b) is consistent with the fact that matrix B (and then B 1 too) is upper and lower bounded. hen it results: 1 0 < B m B B M ( ), If we set, for instance: Bˆ 1 I ˆ BM Bm we obtain: B B I α < 1 B M B m BM Bm Assumption c) concerns centrifugal, Coriolis and gravitational terms. We can take for Φ a uadratic function in the norm of the state: ( ) Φ ξ α0 α1 ξ α ξ

Robust control Let s take now as a control law the following expression: y K K w d P where the new term w will be designed in a way to counteract uncertainty. Letting K [K P K ], we obtain: ξ Hξ η where: H ( H K ) ( w ) 0 I K P K is the matrix, with all eigenvalues in the left half plane, of the closed loop nominal dynamics. If η 0, it is enough to set w 0 in order to return to the nominal case. For the definition of the vector w we proceed with the Lyapunov method.

Robust control Consider the following Lyapunov candidate: V ( ξ) ξ Qξ > 0, ξ 0 where Q is a symmetric positive definite matrix. aking the derivative along the trajectories of the system we obtain: V ξ ξ ξ Qξ ξ ( ) H Q QH ξ ξ Q( η w ) Pξ z Qξ ( η w ) where we have set: z Qξ and we have exploited the fact that, since H has all eigenevalues with negative real parts, whatever positive definite matrix P is chosen, the euation: H Q QH P has only one positive definite solution Q.

Robust control Let s adopt for w the expression: w z ( ξ ) ρ > 0 ρ, z ρ is a positive function of the norm of the state, to be determined. With this choice of w we obtain: z ( η w ) z η ρ( ξ ) z z z z z η ρ( ξ ) ( η ρ( ξ ) If we then guarantee that: ( ξ ) > η,, d z ρ. we obtain that this term, and then V too, is negative.

Robust control Remember that: η 1 ( I B Bˆ ) y B 1 n y K K w d P It follows that: η I B αq < ρ M ( ξ ) α ( d K ξ w ) B K ξ αρ( ξ ) B Φ( ξ ) 1 ˆ 1 B M n We might then select function ρ in such a way that: ρ 1 ( ξ ) ( αq α K ξ B Φ( ξ ) 1 α M M

Robust control Notice that since: Φ ( ξ ) α α ξ α 0 1 ξ in order to satisfy the previous ineuality it will be enough to select: ρ with: ( ξ ) β β ξ β 0 1 ξ β 0 αq M α 1 α 0 B M, β 1 α K α 1 α 1 B M, β α B M 1 α In this way: V ξ Pξ z η ρ, z z ( ξ ) < 0 ξ 0 ξ 0 is a globally asymptotically stable euilibrium state.

Robust control he resulting control scheme is the following one:.. d. d d. Q K K P z ρ( ξ )vers(.) w y ^ B() τ MANIPULAOR.. n(,) ^

Robust control he control law is then composed of three terms: ˆ approximately compensates for the nonlinear terms 1. B( ) y nˆ (, ). K K d P stabilizes the nominal dynamic system in the error z 3. w ρ( ξ ) z gives robustness, counteracting the uncertainty In order to avoid high freuency switching of the control variable (so called chattering) the third term can be approximated as : w ( ξ ) ρ ρ ε ( ξ ) z z z for for z z ε < ε

Adaptive control As an alternative to the robust control, of particular interest when the euations of the dynamic models are reasonably known, but there is uncertainty on the parameters of the model itself, it is possible to design an adaptive control. he adaptive control is based on the linearity of the dynamic model of the manipulator with respect to dynamic parameters: B ( ) C(, ) g( ) Y (,, ) π τ where π is a suitable constant vector of uncertain dynamic parameters (kinematic parameters are assumed to be known without uncertainty). Apparently a techniue based on this relation should make use of measures or estimates of joint accelerations in order to compute the regressor Y, which of course would considerably limit its practical use. he techniue we are going to discuss in the following, elaborated by Slotine and Li, solves the problem without information on the accelerations and without the inversion of the inertia matrix of the manipulator.

Adaptive control Consider first the following control law, where we assume that the dynamic model is known without uncertainty: τ ( ) C(, ) g( ) K σ B r r where K is a positive definite matrix. Let us define: Λ Λ r d r d with Λ positive definite (can be chosen as a diagonal matrix). If we now set: σ Λ r the term K σ corresponds to a P action on the error. Altogether the application of the control law implies: B ( ) σ C(, ) σ K σ 0

Assume as a candidate Lyapunov function the following expression: ( ) ( ) 0 > M B 0 1 1 σ, σ σ σ, V with M positive definite matrix. aking the derivative with respect to time: ( ) ( ) ( ) [ ] ( ) M K M B K C M B B 1, 1 1 V σ σ σ σ σ σ σ σ σ σ where the skew-symmetric property of matrix BC has been exploited. Setting M ΛK we obtain:. 0 Λ Λ < K K, 0 V he state [, σ ] 0 is then a globally asymptotically stable euilibrium. Adaptive control

Consider now a control law based on estimates of parameters: ( ) ( ) ( ) ( ) σ π σ τ r r r r K Y K g C B ˆ,,, ˆ, ˆ ˆ By substituting in the dynamic model: ( ) ( ) ( ) ( ) ( ) ( )π σ σ σ,,,,, r r r r Y g C B K C B Notice that Y does not depend on joint accelerations. where: π π π ˆ ˆ ˆ ˆ g g g C C C B B B Adaptive control

Let s modify the candidate Lyapunov function: ( ) ( ) 0 > π π σ, π π Λ σ σ σ,, 0 1 1 K K B V where K π is a positive definite matrix. aking the derivative with respect to time: If we adopt the following adaptation law of the parameters: ( ) ( ) σ π π r r Τ V Y K K K,,, π Λ Λ from the previous case due to the uncertainty ( )σ π r r Y K,,, ˆ 1 π it results, taking into account that π is constant: K K Λ Λ V hus e asymptotically (and globally) converge to zero.. Adaptive control

Adaptive control From the relation: B ( ) σ C( ) σ K σ Y (,,, )π we obtain that: Y, r r (, r, )( πˆ π) 0, r his does not necessarily imply that ^π tends to π, which depends on the structure of matrix Y. It is actually a direct adaptive control, aimed at finding an effective control law, rather than at identifying the model parameters.

Adaptive control his is the resulting control scheme:.. d. d d. Λ Λ.. r. r σ 1 K π Y. π^ K ^π Y τ MANIP..

Adaptive control We can identify three elements in the control law: 1. Yπˆ can be intepreted as an approximate inverse dynamic control. K σ P-like stabilizing action on the error 3. he vector of the estimates of parameters π is updated following a gradient techniue. Matrix K π determines the speed of convergence of the estimates. Compared to the robust control scheme we have worse performance in the presence of model errors or of partial representation of the dynamic model. On the other hand we have more regular control actions than those obtained with the robust control, which are characterized by potentially dangerous high freuency switching.

Operational space control In the operational space control, the error is directly formed on the operational (Cartesian) space coordinates. he trajectory generated in the operational space is not subject to kinematic inversion. On the other hand the measures of the variables in the operational space are actually the result of direct kinematics computations on the only available measures, i.e. on the joint coordinates measures. We will consider only the case of non-redundant robots not incurring in singular configurations (full rank Jacobian)

P plus gravity compensation As with the joint space, assume that a constant posture x d is assigned. Our goal is to find a control law which ensures the global asymptotic stability of the euilibrium state, using the Lyapunov method. Let s define the error in the operational space: x x d x As a candidate Lyapunov function we take: ( x 1 1, ) B( ) x K x V P By selecting a symmetric positive definite matrix K P, also function V is positive definite.

P plus gravity compensation aking the derivative of functionv: 1 V B ( ) B( ) x K x As x d is constant we have: P x J A ( ) and thus: 1 V B( ) B ( ) J ( ) K x A P 1 ( B ( ) C (, )) ( τ g( ) J ( ) K x A P ). he first term is zero due to the skew-symmetric property of matrix BC.

P plus gravity compensation Consider now the following control law: τ g ( ) J ( ) K x J ( ) K J ( ) A P A A where K is positive definite. We obtain: V ( ) K J ( ) 0 J A A hus the derivative of V is negative semidefinite. With a similar argument as. the one used in joint space, we conclude that the trajectories compatible with V 0 are characterized by: J A ( ) K x 0 P If the Jacobian is full rank we then obtain: x xd x 0 the euilibrium state characterized by zero error (in the operational space) is globally asymptotically stable.

P plus gravity compensation he control law obtained with the Lyapunov method can be interpreted as a P control in the operational space, to which a term is added in the joint space for compensation of gravitational effects: K ẋ J A () K P and K diagonal x d x K P J A () τ MANIPULAOR. x d constant x k(.) g(.)

Inverse dynamics control Consider again the dynamic model of the robot manipulator: B ( ) n(, ) τ where centrifugal, Coriolis, and gravitational terms have been gathered in n, and remember that the inverse dynamics control law: τ B ( ) y n(, ) transforms the system, assuming perfect knowledge of the model, in a system of double integrators: y he problem is now to determine the new input y in such a way to allow tracking of a trajectory x d (t), assigned in the operational space. Notice that taking the derivative of the differential kinematics we have: x d x d ( J (, ) ) J ( ) J (, ) A A A dt dt

Inverse dynamics control he previous relation suggests the following choice for y: y 1 A J By substitution we obtain: x K x K x ( ) ( ) x K x K x J (, ) d P 0 P A x i ( 0) x i hen the error in the operational space is governed by a second order dynamics which can be arbitrarily assigned, on each Cartesian degree of freedom, by suitably selecting the elements of the diagonal matrices K P e K. t Notice that the method needs computation of the inverse of the Jacobian matrix. As such it can be applied only to non-redundant robots in nonsingular configurations.

Inverse dynamics control he inverse dynamics in the operational space can be represented by the following block diagram:.. x d. x d x d x x. K K P B() 1 y J A () τ MANIPULAOR. x k(.) ẋ J A (). J A (,).. η(,)