Predictive functional control of a PUMA robot
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ACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
Predictive functional control of a PUMA robot
A. Vivas, V. Mosquera
Department of Electronic, Instrumentation and Control, University of Cauca
Calle 5 No. 4-70, Popayán, Colombia
avivas@unicauca.edu.co
http://www.ai.ucauca.edu.co
Abstract In the past decade, model predictive control (MPC)
This paper describes an efficient approach for nonlinear has become a good control strategy for a large number of
model predictive control, applied to a 6-DOF arm robot. process [8], [9]. Several works [10], [11], [12] have
The model is first linearized and decoupled by feedback, shown that model based predictive control could be of a
secondly a model predictive control scheme, great interest for handling constrained processes by
implemented with an optimized dynamic model and optimizing, over some manipulated inputs, forecasts of
running within small sampling period, is exhibited. Major process behavior. It provides good performances in term
simulation results performed using numerical values of of rapidity, disturbances and errors cancellations.
an industrial PUMA 560 robot prove the effectiveness of However, a significant number of industrial applications
the proposed approach. The nonlinear model-based can be found, for instance, in chemicals industries or
predictive control and the widely used computed torque food processing that are processes with relatively slow
control are compared. Tracking performance and dynamics. But very few results concern the computation
robustness with respect to external disturbances or errors of the model predictive control with nonlinear and high
in the model are enlightened. dynamic process such as robot links [13], [14], where the
model is highly nonlinear and the digital control has to be
performed within only few milliseconds.
Keywords: Robot control, dynamic modelisation,
computed torque control, predictive control. An special predictive strategy is presented in this
study (predictive functional control [15], [16]), and it will
1. Introduction be compared with the computed torque control. The robot
platform used to test these controllers is a PUMA 560
robot, widely used in robotics research, for which there is
Many strategies have been used in the last years for a substantial literature. Test will remark the performances
robot control [1], [2], [3]. In industrial and commercial in tracking and robustness behavior.
robot arms, controllers are still usually PID. But robotic
manipulators have many serially linked components, the This paper is organized as follows: Section (2)
manipulator dynamics is highly nonlinear with strong resumes the dynamic model formulation of a robot,
couplings existing between joints, that complicate the Section (3) details the model predictive functional control,
task of a simple PID, especially if the velocities and Section (4) is dedicated to define the model based control
accelerations of the robot are high. Better solutions are strategies of this study, Section (5) list major simulation
then implemented with model based controllers, that use results in terms of tracking performance and robustness.
a mathematical model of the robot to explicitly Finally, conclusions are given in Section (6).
compensate the dynamic terms. The most common
strategy of model based control is the computed torque
control [4], [5], widely used for industrial robot arms.
This strategy is relatively easy to implement but the 2. Dynamic model formulation
uncertainties presents in the model difficult to design an
effective control algorithm based on an exact It is well known that the behavior of an n-DOF rigid-
mathematical model. Other solutions may be then link arm is governed by the following equation:
implemented to give to the system the robustness needed
[6], [7]. Γ = A(q )q+ C (q, q )q + g (q ) (1)ACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
Where Γ is the actuation torque, A(q ) is a symmetric Internal Modeling: The model used is a linear one given
and positive definite inertia matrix, C (q, q ) is the matrix by:
of the Coriolis and centrifugal forces, and g(q) is the
xM (n) = FM xM (n −1) + GM u(n −1)
gravity force. This robot dynamic equation can also be
written in a compact form: yM ( n) = C M T x M ( n) (5)
Γ = A(q ) q+ H (q , q ) (2) where xM is the state, u is the input, yM is the measured
model output, FM, GM and CM are respectively matrices
where H includes centrifugal, Coriolis and gravitational or vectors of the right dimension.
forces.
If the system is unstable, the problem of robustness
The dynamic terms in Eq.(2) are highly nonlinear and caused by the controller's cancellation of the poles is
coupled. The first step is to design a linearizer to usually solved by a model decomposition [15].
dynamically linearize and decouple the robot dynamic
model. The possibility of finding such a linearizing Reference trajectory: The predictive control strategy of
controller is guaranteed by the particular form of system the MPC is summarized in Figure 1. Given the set point
dynamic. In fact, Eq.(2) is linear in the control Γ and has trajectory over a receding horizon [0, h], the predicted
a full-rank matrix A(q) which can be inverted for any process output yˆ P will reach the future set point
manipulator configuration. following a reference trajectory yR .
Thus, the following linearization is proposed:
In Figure 1, ε (n) = c(n) − yP (n) is the position
Γ = Aˆ (q ) u + Hˆ (q , q ) (3) tracking error at time n, c is the set point trajectory, yP is
the process output, and CLTR is the closed loop time
response.
where u represents a new input vector. Â and Ĥ are
estimations of the real manipulator dynamics. In the Over the receding horizon, the reference trajectory
absence of disturbance and when the dynamic model is
yR , which is the path towards the future set point, is
perfectly known, Â = A and Ĥ = H . In this case u has
given by:
the form of a joint acceleration:
c(n + i ) − yR (n + i ) = α i (c(n) − yP ( n)) for 0 ≤ i ≤ h (6)
u = q (4)
where α (0< αACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
nh nB
{ }
2
D(n) = ∑ yˆ P (n + h j ) − yR (n + h j ) (7) yF (n + i ) = ∑ µk (n) yBK (i ) 0≤i ≤ k (15)
j =1 k =1
where nh is the number of coincidence time point, hj are where y BK is the model output response to uBK .
the coincidence time points over the prediction horizon. Assuming that the predicted future output error is
The predicted output yˆ P is usually defined as: approximated by a polynomial function, it follows:
de
yˆ P (n + 1) = yM (n + i ) + eˆ(n + i ) 1≤ i ≤ h (8)
eˆ(n + i ) = e(n) + ∑ em (n)i m for 1 ≤ i ≤ h (16)
m =1
where yM is the model output and ê is the predicted
future output error. where de is the degree of the polynomial approximation
of the error, em are coefficients computed on-line
It may be convenient to add a smoothing control term knowing the past and present output error.
in the performance index. The index becomes:
The minimization of the performance index without
nh smoothing control term, in the case of the polynomial
{ }
2
D(n) = ∑ yˆ P (n + h j ) − yR (n + h j ) + λ {u (n) − u (n − 1)} (9) base functions, leads to the applied control variable:
2
j =1
u (n) = k0 {c( n) − yP (n)}
where u is the control variable.
max( dc , de )
Control variable: The future control variable is assumed
+ ∑ km {cm (n) − em (n)} + VX T xM (n) (17)
m =1
to be composed of a priori known functions:
where dc is the degree of the polynomial approximation
nB
u (n + i ) = ∑ µ k (n)u BK (i ) 0 ≤ i ≤ h (10) of the set point and the gains k0 , km , VX T are computed
k =1 off-line [15].
where µk are the coefficients to be computed during the Therefore the control variable is composed of three
optimization of the performance index, uBK are the base terms: the first one is due to the tracking position error,
functions of the control sequence, nB is the number of the second one is placed especially for disturbance
base functions. rejection and the last one corresponds to a model
compensation.
The choice of the base functions depends on the
nature of the set point and the process. These base
functions will be represented by:
4. Control strategies
Description of the PUMA robot: The PUMA 560
u BK (i ) = i k −1 ∀k (11) (Programmable Universal Machine for Assembly)
depicted in Figure 2 is a 6 DOF manipulator with
In fact, only the first term is effectively applied for revolute joints that has been widely used in industry and
the control, that is: still is in academia. The PUMA uses DC servo motors as
its actuators, has high resolution relative encoders for
nB accurate positioning and potentiometers for calibration.
u (n) = ∑ µ k (n)u BK (0) (12)
k =1
The model output is composed of two parts:
yM (n + i ) = yUF (n + i ) + yF (n + i ) 1 ≤ i ≤ h (13)
where yUF is the free (unforced) output response (u = 0),
y F is the forced output response to the control variable
given by Equation (10).
Given Equation (6) and Equation (10), it follows:
yUF (n + i ) = CM T FM xM (n) 0≤i ≤ h (14)
Figure 2. PUMA 560 robotACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
Implementation of inverse dynamics control laws The resulting scheme is shown in Figure 3, in which
indeed requires that parameters of the system dynamic two feedback loops are represented; an inner loop based
model are accurately known. To simplify the dynamic on the manipulator dynamic model and a outer loop
model, some parameters are grouping or eliminated to operating on the tracking error. The function of the inner
obtain a set of base parameters for the PUMA robot [5], loop is to obtain a linear and decoupled input/output
show in Table 1 and Table 2. In these tables, the term relationship, whereas the outer loop is required to
“G” denotes the parameters that includes other terms. stabilize the overall system.
An identification procedure gives the parameter Predictive functional control: The MPC algorithm is
values show in Table 3 [18]. These estimated values will implemented with an internal model composed of a
be considered as the nominal values during the double set of integrators, that represents the non-linear
simulations. and decoupled system. To facilitate the stabilization
process, two feedback loops are added to the internal
Computed torque control: Assuming that only the model: a position and a velocity loop, with gains Kp = 2
motion is specified as desired position qd, the computed and Kv = 7 respectively.
torque control (CTC) computes the required input control
as follows: Three different base functions are used: step, ramp
and parabola. The close loop response time (CLTR) is
w = K p (q d − q ) − K v q (18) fixed to 32*Tsampling in order to ensure the compromise
between the tracking performances and the disturbance
rejection. Three coincidence time points on the prediction
where Kp and Kv are the controller gains. The tuning of horizon are defined and the predicted future output error
CTC controller for the PUMA robot leads to the gains is approximated by a polynomial of degree 2. The
show in Table 4. smoothing control term is fixed at 0.35. PFC scheme is
shown in Figure 4.
Table 1. Inertia tensor parameters
j XXj XYj XZj YYj YZj ZZj
1 0 0 0 0 0 ZZG1 Table 4. Kp and Kv gains
2 XXG2 0 0 0 0 ZZG2 1 2 3 4 5 6
3 XXG3 0 0 0 0 ZZG3 Kp 18000 10000 8900 6200 10500 5500
4 XXG4 0 0 0 0 ZZG4 Kv 19 16 16 14 18 16
5 XXG5 0 0 0 0 ZZG5
6 XXG6 0 0 0 0 ZZ6
Table 2. First moment of inertia, mass and motor inertia parameters
Kp
j MXj MYj MZj Mj Iaj Non-linear compensation and
decoupling
1 0 0 0 0 0 q d
u Aˆ (q)
Γ
2 MXG2 MY2 0 0 0 q d Kv Robot
qd
3 0 MYG3 0 0 Ia3
4 0 0 0 0 Ia4 q q
Ki ∫ Hˆ (q, q) q
5 0 MYG5 0 0 Ia5
6 0 0 0 0 Ia6 Stabilizing linear
control
Table 3. Parameters of the PUMA robot Figure 3. Joint inverse dynamics control
Parameter Values Parameter Values
XXG2 -1.8566 ZZ6 5.8e-6
XXG3 0.3069 MXG2 5.3086
XXG4 6.0e-4 MY2 0.1120
XXG5 1.2e-4 MYG3 0.3420 Kp Non-linear compensation and
XXG6 0.0 MYG5 0.0012 decoupling
q
ZZG1 2.4441 IA3 200e-6 qd
PFC
+ - Γ
Aˆ (q) Robot
ZZG2 2.2558 IA4 3.3e-5
ZZG3 0.3150 IA5 3.3e-5 -
Stabilizing linear q
ZZG4 0.0039 IA6 3.3e-5 control Hˆ (q,q)
Kv
ZZG5 6.9e-6 q
Note: the units for the inertia tensor parameters are
Kg.m2, for the first moment of inertia Kg.m, and for the Figure 4. Predictive Functional Control
motor inertia Kg.m2.ACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
5. Simulation results 3
2
Computed torque control and predictive functional
control are implemented in simulation environment 1
(Matlab/Simulink). Tracking performance and 0
Control torque (N.m)
robustness (disturbance rejection and errors in the model) -1
are compared. -2
-3
Tracking performances: These tests are running within 1
msecond sampling period. A fifth order trajectories with -4
trapezoidal velocity profile are designed as set point for -5
each motor, being the final positions 1.2, 0.7, 0.5, 0.3, -6
0.65, and 0.35 radians respectively. Figures 5 and 6 show
-7
tracking error for CTC and PFC control respectively, and 0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (ms)
Figures 7 and 8 show their control torques.
-3 Figure 8. Control torque for PFC
x 10
3
Tracking performances are very advantageous for the
2.5
predictive control. In the same way, control torque is a
2
little smaller for PFC due to its smoothing control term.
Tracking error (rad)
1.5 Disturbance rejection: An unknown output disturbance
is applied at 0.3 seconds. It is constant and equal to 0.2
1 radians for all motors. Figure 9 show disturbance
rejection for CTC and Figure 10 for PFC control.
0.5
0
With the same controller tuning, PFC control is more
robust than CTC control with respect to external
-0.5 disturbances and also the time response to the
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (ms) disturbances is better.
Figure 5. Tracking error for CTC 0.25
-3
x 10
3 0.2
0.15
2.5
0.1
Disturbance rejection (rad)
2 0.05
Tracking error (rad)
0
1.5
-0.05
1
-0.1
0.5 -0.15
-0.2
0
-0.25
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (ms)
-0.5
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (ms) Figure 9. Disturbance rejection for CTC
Figure 6. Tracking error for PFC
0.25
3
0.2
2
0.15
1
0.1
Disturbance rejection (rad)
0
Control torque (N.m)
0.05
-1
0
-2
-0.05
-3
-0.1
-4
-0.15
-5
-0.2
-6
-0.25
-7 0 200 400 600 800 1000 1200 1400 1600 1800 2000
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time (ms)
Time (ms)
Figure 7. Control torque for CTC Figure 10. Disturbance rejection for PFCACSE 05 Conference, 19-21 December 2005, CICC, Cairo, Egypt
Errors in the model: The model mismatch is obtained 7. References
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1613, 1994.You can also read
