Introduction to centralized control
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1 ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino
2 Control Part 2
3 Introduction to centralized control Independent joint decentralized control may prove inadequate when the user requires high task velocities structured disturbance torques (Coriolis, centrifugal) greatly influence the robot behavior motors are of direct drive type since non gearboxes are present, their beneficial effect is absent and nonlinear effects and coupling effects become important In these cases the disturbances torques may cause large errors on the reference trajectory tracking Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 3
4 Introduction to centralized control Since in these cases it is not possible to sufficiently reduce the effects due to the disturbance torques, it is more convenient to try to cancel such torques, adopting control algorithms that use nonlinear compensation terms We call this architecture centralized since the applied joint command torques are function also of the other joint positions and velocities. The approach is not local anymore, as happens in the decentralized architecture, where each joint controller uses only the local joint information (position and velocity) Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 4
5 Centralized control In the centralized control architectures the robot is considered as a MIMO system, with n inputs (the joint torques) and n outputs (the joint positions ) that interact with each other according to the nonlinear dynamic equations of the robot model The centralized control algorithms shall take into account this dynamic model, and usually they have a nonlinear form The main centralized control architecture is called inverse dynamics, since the command torques are computed from the robot dynamic equation and from the knowledge of the joint variables (positions and velocities), i.e., as a solution of an inverse dynamic problem Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 5
6 Inverse dynamics control Consider again the following simplified dynamic model of the robot Mqq ( ) ɺɺ hqq (, ɺ) = uc We can compute a control command as a function of the dynamic model, where the inertia and disturbance terms are approximate Reference acceleration h ɵ qɺɺ r M u c ROBOT qɺ q a c Additional acceleration command signal u M c ( q ɺ a r c ) h ɵ Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 6
7 Approximate linearization Hence, developing the previous equation, we have: Mqq ( ) ɺɺ hqq (, ɺ) = M ( qɺɺ a) h ɵ ( ) ( ) qɺɺ a M q hqqɺ h ɵ r c qɺɺ = M 1 () qm 1 () (, ) r c 1 M ( qm ) = I Eq () 1 Eq ( ) = M ( qm ) I Here we try to invert the inertia matrix hqq (, ) h ɵ = hqq (, ɺ) ( ɺ ) Here we try to send to zero the disturbance terms Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 7
8 Conclusions qɺɺ ( qqɺɺɺ, q a) = qɺɺ a) η,, ( r c r c Structured disturbance If we can cancel this part, the system becomes linear and decoupled, but unstable η ( ) 1 qqq ɺɺɺ a = Eq qɺɺ a,,, ( )( ) M ( q) hq (, qɺ) r c r c Approximation in inertia model Approximation in Coriolis model Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 8
9 In general If we are able to compute the exact model of the nonlinear dynamical system, we can build the following control architecture, where a c is a suitable acceleration control signal, whose definition will allow to obtain asymptotic stability and other performances a c ( ) M q u ROBOT q qɺ hqqɺ (, ) Basilio Bona - DAUIN -PoliTo ROBOTICS 01PEEQW 9
10 Exact linearization and decoupling The control scheme above performs an exact system linearization a c The resulting system is decoupled: it consists in n double integrators; the i-th component a c,i of the new acceleration signal influences only the behavior of the i-th joint component q c,i that is independent of the other joints motion M( q ) u ROBOT hqqɺ (, ) q qɺ a c qɺ q 10
11 State variable representation 11
12 Error variable representation This term is equivalent to the injection into the system of a structured nonlinear disturbance that can make it unstable in spite of the control design 12
13 Controller design 13
14 Controller design qɺɺ r h ɵ M M, h u c Nonlinear Inner Loop ROBOT qɺ q a c Linear Outer Loop 14
15 Inner loop outer loop (nonlinear linearizing control) 15
16 Exact linearization 16
17 Exact linearization hqqɺ (, ) qɺɺ r M( q) u c ROBOT M, h qɺ q a c Inner Loop qɺɺ r a c 1 s 1 s qɺ 1 qɺ n 1 s 1 s q 1 q n Control design 17
18 PD outer loop control design 18
19 PD outer loop control design 19
20 PD outer loop control design Inner Loop hqqɺ (, ) qɺɺ r M( q) u c ROBOT M, h qɺ q a c Outer Loop qɺ r q r K D K P 20
21 PID outer loop control design 21
22 PID outer loop control design Inner Loop h qɺɺ r a c M u c ROBOT qɺ q Outer Loop a PID controller for the outer loop qɺ r q r K D K P K I s 22
23 PID outer loop control design 23
24 Practical aspects of exact linearization Exact linearization hypothesis implies the capacity to compute on-line the model matrices Dynamic models must be perfectly known, without errors or approximations The model matrices must be computed online; at every sampling time (approx. 1 ms) the inverse dynamics equations must be solved. Software and hardware architectures must be able to do so Unmodeled dynamics (e.g., elastic vibrations) are not taken into account In practice it is impossible to satisfy all these assumptions at the same time 24
25 Approximate linearization In practice one may adopt a control scheme based on the imperfect compensation of the robot inverse dynamics There are different possibilities to do so: 1. Only a part of the robot dynamics is computed and used in the controller (usually the dominant one and/or that is best known), leaving to the outer loop the task of guaranteeing the overall stability and the reference trajectory tracking 2. A feedforward compensation of the robot dynamics; it uses the reference values of the desired trajectory instead of the real measured joint variables and velocities to build the model matrices 25
26 Approximate linearization 3. Robust control techniques; these are advanced techniques that allow to overcome the effects of the approximations and other errors introduced by the real-time computation of the inverse dynamics 4. Adaptive control techniques; they provide an online estimation of the true model parameters that are successively used in the controller Also for the outer loop controller it is possible to adopt more complex algorithms than the simple PD controller. A PID controller will be able to cancel the steady state effects of any additive constant disturbance that the inner loop cannot cancel 26
27 Approximate linearization 27
28 Approximate linearization Under the assumption that only an approximate compensation is possible, the external control law becomes: Where u = Mqa ( ) hqq ɵ (, ɺ ) c M M; h ɵ h represents an estimate of the true matrices. These matrices could be both the best available approximation of the true matrices, or the result of an a-priori decision that simplifies the model c 28
29 Approximate linearization Inner loop h ɵ qɺɺ r a c M u c ROBOT qɺ q We assume to use a PD controller for the outer loop 29
30 Approximate linearization There are some inverse dynamics architectures with approximate linearization, that include PD control with gravity compensation Independent joint control Inverse dynamics feedforward control (also called computed torque method) 30
31 PD control with gravity compensation Exact knowledge of the gravity terms 31
32 PD control with gravity compensation Inner loop τ ( q ) g u c ROBOT qɺ q a c Outer loop qɺ r q r eɺ e K D K P 32
33 PD control with gravity compensation (A) 33
34 PD control with gravity compensation 34
35 PD control with gravity compensation 35
36 PD control with gravity compensation 36
37 PD control with gravity compensation 37
38 Independent joint control 38
39 Independent joint control 39
40 Independent joint control Inner loop u = a c c ROBOT qɺ q Outer loop qɺ r q r eɺ e K D K P 40
41 Computed Torque Method The computed torque method represents an solution midway between the decentralized control and the inverse dynamics control It is mainly used when the reduced processing power cannot afford the complete inverse dynamics implementation When the real-time constraints are strict, the feedforward term can be approximated Considering only the diagonal terms of the inertia matrix and the gravitational effects, that are dominant when the velocities are small Computing off-line the feedforward torque terms and storing them in a mass memory or look-up table; this is possible when cyclic trajectories are performed again and again 41
42 Computed Torque Method 42
43 Computed Torque Method 43
44 Computed Torque Method qɺɺ r a c u c ROBOT qɺ q Outer loop ɺ r eɺ K D q r e K P 44
45 References 45
46 Feedback linearization the general framework 46
47 Feedback linearization the general framework 47
48 Feedback linearization the general framework 48
49 Feedback linearization the general framework 49
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