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24URAL MATHEMATICAL JOURNAL, Vol. 5, No. 2, 2019, pp. 13-20
DOI: 10.15826/umj.2019.2.002
IMPULSE CONTROL OF THE MANIPULATION ROBOT
Yurii F. Dolgii1'2'", Alexander N. Sesekin1'2'"", Ilya A. Chupin1'"""
1Ural Federal University, 19 Mira str., Ekaterinburg, 620002, Russia
2Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences,
16 S. Kovalevskaya Str., Ekaterinburg, 620990, Russia
"[email protected], ""[email protected], """[email protected]
Abstract: A nonlinear control problem for a manipulation robot is considered. The solvability conditions for the problem are obtained in the class of special impulse controls. To achieve the control goal, the kinetic energy of the manipulation robot is used. When finding analytical formulas for controls, the classical first integrals of Lagrangian mechanics were used. The effectiveness of the proposed algorithm is illustrated by computer simulation.
Keywords: Manipulation robot, Impulse controls, First integrals.
Introduction
The purpose of controlling the manipulation robot is to transfer it from the initial position to the final. The significant nonlinearity of the mathematical model describing the movements of the manipulation robot does not allow the use of the methods of the mathematical control theory directly for the original model. Decomposition methods make it possible to reduce the dimension of the control problem, passing to approximate linear or integrable mathematical control models [1, 2]. The work considers a manipulation robot with three degrees of freedom, imitating the movement of a human hand, described in the monograph [3]. In [4], the problem of controlling the horizontal two-dimensional motion of this robot was studied. The original nonlinear control model was replaced with an integrable controlled model. The Pontryagin maximum principle was used to find controls in the performance problem. The use of special impulse controls in this work allows the kinetic energy of the manipulation robot to be used to reach the final position. This approach preserves the integrability of the original controlled mathematical model and, when solving the problem, uses the classical first integrals of Lagrangian mechanics. The article concludes with the results of numerical simulation of the algorithm.
1. Mathematical statement of the problem
A manipulation robot with three degrees of freedom, imitating the movement of a human hand, described in the monograph [3, p. 263]. Figure 1 shows a robot manipulator. The number 1 indicates the base of the robotic arm, 2 is a rack of vertically oriented shaft. This shaft is rigidly connected to the guide beam 4 and hand 5.
Kinetic energy of the manipulation robot is determined by the formula
T = - (,7ih~/2 + (Ji + .hW2 + m,2(zr2 + x'2 + xV2))
where x is the coordinate of the center of mass of the arm, x > 0; z is the height of the arm, z > 0; p is the angle of rotation of the hand; mi is the total mass of the shaft that rotates the arm, and the guides that implement the horizontal displacement of the arm; m2 is the mass of the arm; J1 is the moment of inertia of the shaft and guides relative to the vertical axis; J2 is the central moment of inertia of the arm relative to the vertical axis. The potential energy of gravity is V = (m1 + m2)g, where g is the gravitational acceleration.
The second-order Lagrange equations for the mechanical system under consideration are of the form [3, p. 263]
z" + g = ui, (1.1)
x" - xp'2 = U2, (1.2)
p'' + a2 (x2p')' = U3. (1.3)
Here u1 = P/(m1 + m2), u2 = F/m2, u3 = M/(J1 + J2) are the control force actions, where P is the magnitude of the longitudinal force acting along the vertical axis, F is the magnitude of the longitudinal force acting along the horizontal guides, M is the magnitude of the moment of force directed along the vertical axis, a = -\/m2/(Ji + J2).
It is required to find the controls u1, u2, u3 that translate the system (1.1)-(1.3) from the initial equilibrium position (z0, xo, po)T, z0 > 0, xo > 0, to the given end position (z*, x*, p*)T, z* > 0, x* > 0, z* = zo, x* = xo, p* = po.
To solve this problem, we use a special set U of impulse controls defined by the formulas
u1(t) = zoS(t), u2(t) = xoS(t), u3(t) = poS(t), t € R,
where ¿(-) is the Dirac impulse function. Impulse controls at the initial moment of time t = 0 to the dynamic system (1.1)-(1.3), which are in equilibrium, report the initial velocities z'(0) = zo, x'(0) = xo, p'(0) = po. The initial speeds are control parameters, the choice of which should ensure that the dynamic system (1.1)-(1.3) falls into the final position.
2. Impulse control parameters
The problem is divided into two problems. In the first problem, for the equation (1.1) that determines the vertical movement of the manipulation robot, we use the set U1 of impulse controls defined by the formula u1(t) = zo^(t), t € R. It is required to find the control u1, which transfers the dynamical system (1.1) from the initial equilibrium position zo, zo > 0, to the given final
position z*, z* > 0, z* = z0. In the second problem, for the system of equations (1.2), (1.3), which determines the horizontal movement of the manipulation robot, we use the set U2 of impulse controls defined by the formulas u2(t) = x0^(t), u3(t) = (0^(t), t € R. It is required to find controls u2,u3 that translate the dynamical system (1.2), (1.3) from the initial equilibrium position (x0, (0)T , x0 > 0, to the given end position (x*, (*)T , x* > 0, x* = x0, (* = (0.
Lemma 1. In the set Ui there are impulse controls ui that transfer the dynamical system (1.1) from an arbitrary initial position z0, z0 > 0, to an arbitrary end position z*, z* > 0, z* = z0.
Proof. The impulse control u1 € U1 provides the dynamic system (1.1) at the initial moment t = 0 the initial velocity z'(0) = z0. For t > 0, the motion of a free dynamic system is determined by the differential equation z'' + g = 0 with the initial conditions z(0) = z0, z'(0) = z0. The vertical movement is described by the formula z(t) = —gt2/2 + z01 + z0, t > 0. For any t1 > 0 there is a unique value of the control parameter z0 for which the equality z(r1) = z*, z0 = t-1 (z* — z0)+g^1/2 is true.
We show that the velocity at the finite moment of time is minimal in absolute value if t\ = \/2\z* — zo|/g. The velocity at the finite moment of time is determined by the formula
z'(T1) = — gT1 + z0 = —gT1/2 + t-1 (z* — z0).
If z* > zo > 0, then we have z'(ti) = 0 under the condition t\ = \/2(z* — zo)/g■ The function /(r) = gi/2 + r_1 (zo — z*) has a minimum at r = t\ = \J'2 (zo — z*) /g which is eciual to ■s/2g (z0 - z*) if z0 > z* > 0.
The impulse controls u2, u3 € U2 prescribe the dynamic system (1.2), (1.3) at the initial moment t = 0 the initial speeds x'(0) = x0, (0) = (0. For t > 0 the motion of a free dynamic system is determined by differential equations x'' — x('2 = 0, ('' + a2 (x2() = 0 with initial conditions x(0) = x0, ((0) = (0, x'(0) = x0, (0) = ((0. For the horizontal free movement of the manipulation robot, kinetic energy and momentum are kept the same [5]
T-2 = \ ((Ji + J2W2 + m2(xn2 + xV2)) = const, dJ2
P'2 = TTT = (^1 + J'2 + m2X2)<p' = const. d('
When describing horizontal motion for t > 0, we replace the system (1.1), (1.2) with the system of differential equations
a2x'2 + (1 + a2 x2)('2 = C1, (2.1)
(1 + a2x2)(' = C2, (2.2)
where
C1 = a2 xc 0 + (1 + a2 x2)(0, C2 = (1 + a2x0)(0.
Lemma 2. Let the conditions
. . f \/l + d2xMs
IV* " Vol < / . f—ff' 0 < Xo <
J \/(1 + a2s2)(s2 - x0)
xo
x° /-;-r
. . f J1 + a2x2ds
□
(2.3)
(2.4)
hold. Then in the set U2 there are impulse controls u1, u2 that move the dynamical system (1.2), (1.3) from the starting position (xo, po)T , xo > 0, to the ending position (x*, p*)T , x* > 0, x* = xo, p* = p0.
Proof. Under the condition xo = 0, the system of differential equations (2.1), (2.2) is transformed to the following form
x2_x2
Sgn ¿0 \l ¿0 + (1 + a'2x0) <¿>0 -,' , 2 °2 ' (2-5)
y 1 | aa X
1 + a2x2
where
2 2
X = {x € R+ : + (1 + a2xl)pl * > o). L 1 + a2 x2 J
To move the motion of the dynamical system (2.5), (2.6) from the initial to the final position, the control parameters Xo, po must satisfy the conditions
p0 = 0, sgn Xo = sgn(x* - Xo), sgn po = sgn(p* - po).
We introduce the parameter p = |xo|/|po|. Now the description of the set X = X(p) is simplified. As a result, we have
X(p) = R+ under p > xq \J 1 + a2.^,
X(p) = R+/(0, xi(p)) under p < xq \Jl + a2.^,
where
^ = xj(l + a2xj) p2 + V 1 + a2xo
The equation (2.5) is converted to
22 «X/ «^Pl
a;' = sgn (a:* - x0)\ip0\\ p2 + (1 + a2x20) ——xeX. (2.7)
y o 1 + a2x2
It is also valid for p = 0. Using (2.7) and (2.6), we obtain a differential equation for finding the trajectory of a horizontal movement
dx sgn(x — xo)(1 + a2x2) / „ „ 0. x2 — xo „„/ x
dp sgn(p* - po)(1 + a2xg) y o 1 + a2x2
Integrating the differential equation, we find the equation of the trajectory of a horizontal movement
x 2 2
sgn (.t* - .To) f-(l + ax0)d,s ^ ^ = ggn ^ _ ^^ _ ^ ^ ^ ^^
xo (1 + a2s2) Jp2 + (1 + a2xl) S
V 1 + a2s2
The trajectory passes through the end point if the condition is true f (1 + a2xo)ds
•to (1 + a2s2)\lp2 + (1 + a2x2)^^ V 1 + a2s2
= |p* - po|, x* € X(p).
x
We select the value of the parameter p to satisfy the condition obtained. For 0 < x0 < x*, the required value of the parameter p is determined by the equation
ds
s2 — x2
\(f>* ~ (f>o\
1 + a2Xn :
0 < p <
•to (1 + a2s2)\ p2 + (1 + q;2Xq)-^--PV
V 1 + a2s2
For 0 < x* < x0, the required value of the parameter p is determined by the equation
(2.8)
ds
|<£* - ^o |
(1 + a2s2)\/p2 + (1 + Q2Xq) S X2°, V 1 + a2s2
2 1 \ aa Xt
2> Pkp<P<+ OO,
(2.9)
where
pfcp
(1 + Q2Xq)(XQ -x2)
1 + a2x2
The equation (2.8) has a unique solution p = p* under the condition (2.3) and equation (2.9) also has a unique solution p = p* if the conditions (2.4) hold.
Suppose that the condition (2.3) holds for 0 < x0 < x* and for 0 < x* < x0 the condition (2.4) is satisfied. Then impulse control u2(t) = x05(t), u3(i) = ^>0¿(t) (t € R) bring the dynamic system (1.2), (1.3) to the given final position, if the control parameters are determined by the formulas
x0 = |x01 sgn (x* - x0), ^0 = |^0| sgn(^* - ^0), |xx01 = p*|^0|,
where p* is the root of the equation (2.8) for 0 < x0 < x*, and for 0 < x* < x0 there is a root of the equation (2.9).
Integrating the differential equation (2.7), we find the arrival time of the motion of the dynamical system (1.2), (1.3) at the end point
1
T2 =
№ 0|
ds
p* + (1 + a2 x0)
22 s - x0
1 + a2s2
We synchronize the arrival times of the movements of the dynamical systems (1.1)-(1.3) to the end points. □
Theorem 1. Let the conditions of Lemma 2 be satisfied. Then the values of the parameters of the impulse controls that move the dynamical system (1.1)—(1.3) from the initial position (z0,x0,^0)T, Z0,x0 > 0, to end position (z*,x*,^>*)T, z* > 0, x* > 0, z* = Z0, x* = x0, = ^0 are defined by formulas
z0 = s/Zgiz* ~ 20) for 0 < z0 < z*,
Z0 = 0 for 0 < z* < z0,
y/gsgn(<p* -Ifio)
x 0
y/2\z* - -sol
vfgy* \/2|z* - ZqI
ds
22 s - x0
I
0
x
*
Proof. Using Lemma 1 and the arrival time t1 of the motion of the dynamical system (1.1) to the end point, we find the value of the control parameter zo. The synchronization condition t1 = t2 of the arrival times of the motions of dynamical systems (1.1)—(1.3) at end points determines the arrival time r = y/2\z* — zo\/g of dynamic system movements (1.1)—(1.3) to the end point (z*,x*, p*) and the equation for the control parameter po. From this equation we find
w = ^
s/2\2* - -o0|
ds
•T0 \Jrt + (l + (*2x2o)*+ Js°2
Using Lemma 2, we find the control parameters po, xo. □
3. Stabilization of manipulation robot in a final position
When stabilizing the manipulation robot in a small neighborhood of the final position, we use special positional controls, the choice of which turns the final position into a stable equilibrium position of the controlled system. For this purpose we use substitutions for coordinates
z = z* + i, x = x* + x, p = p* + p
and controls
u1 = u1 + g, u2 = u2, u3 = u3(1 + a2x2).
In a small neighborhood of the final equilibrium, the controlled system (1.1)-(1.3) is replaced by the following controlled system
z'' = u1, x'' = u2, p'' = u3. (3.1)
We find the stabilizing control using the theory of optimal stabilization for linear systems with quadratic quality criteria. Choosing the quality criterion
J1 = J (?2(t) + k2?'2(t) + u?(t)) dt, k1 > 0, (3.2)
o
for the first control of the system (3.1), we find the stabilizing control
u1 = —z — y k2 + 2z'.
We also can find stabilizing controls for the second and third equations in (3.1)
u2 = —x — y + 2x', u3 = — p — y fc2 + 2p?',
using a quality criteria similar to (3.2) with constants k2 and k2, respectively.
4. Numerical modeling
In the numerical simulation of the system motions (1.1)-(1.3), the following values of the parameters of the mechanical system were used
x
*
m-1 = 20, m-2 = 8, J1 = 12, J2 = 6, g = 9.8.
The start and the end positions are zo = 0, xo = 0, = 0 and z* = 1.4, x* = 0.5, = 1.2. We take the controls
■Ui (t,z) = uf (t) + u1s(z), u2(t,x) = uf (t) + u2s(x), u3(t, = uf (t) + ufs (^>). The program control is defined by formulas
uf (t) = zo^(t), u2r(t) = xo5(t), uf(t) = <^(t), t € R, where the parameters are given by formulas
z0 = \/29z*i =
p*<^ o, o
1 + a2s2
2I j V p2(l +a2s2) + s2
ds.
Here p = p* is the positive root of the equation
ds
a/(1 + a2s2)((l + a2s2)p2 + s2)
= <£*
Impulse controls moves the mechanical system into equilibrium, the initial speeds are
z' (+0) = zo, x' (+0) = xo, <(+0) = <o.
We also consider software controls in the form of rectangular impulses, which are approximations of ideal impulses
uf (t) = zo¿A(i), u2r(t) = Xo¿A(i), u3r(t) = <o^A(t), t e R,
where
¿A(t) = 1/A, t e (0, A), ¿A(t) =0, t e R/(0, A), A = 0.1.
For these controls, the initial velocities of the equilibrium mechanical system are determined by the formulas z'(0) = 0, x'(0) = 0, <¿(0) = 0.
Positional controls are determined by the following formulas
ups(z) = 0, f (z) = g - (z - z*) - yfcf + 2,
u
ups(x) = 0,
u
ps
(x) = —(x — x*) — ^/kl + 2 x',
up%) = 0,
0 < z < z* — e1, z > z* — ei, 0 < x < x* — e2, x > x* — e2, 0 < ^ < <£* — e3,
u
ps
(ip) = -(1 + a2x2)((tp - ip*) + ^k'i + 2 Lp'), tp> e3,
where
ki = 1, k2 = 1, fcs = 1, ei =0.1, e2 = 0.1, £3 = 0.1. In the final position, the following conditions must be met
|z(Tf)| < e, |x(Tf)| < e, |<(r/)| < e, |z'(t/)| < e, |x'(t/)| < e, <(t/)| < e.
In the computational experiment, we assumed that e = 0.01.
The time movement to the final position is t/ = 7.5 sec. for impulse controls and t/ = 7.51 sec. for rectangular impulses approximating ideal impulse actions. Projections of phase trajectories on state planes (x,x') and (<,<') are shown in Fig. 2 and Fig. 3.
The trajectories corresponding to impulse controls are shown in blue, the trajectories corresponding to the approximations of ideal impulse controls are shown in brown.
x
*
2
3
x 1.2 -
1.0
0.8
0.6
0.4
0.2
-0.2
¥
2.0 1.5 1.0 0.5
Figure 3. State plane (p, p'). 5. Conclusion
The impulse control is constructed in the work that transfers the manipulator from a given position to its final position. A computational experiment showing the efficiency of the proposed algorithm is presented. The proposed algorithm is simulated in the case when the ideal impulse is approximated by the usual bounded control.
REFERENCES
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2. Chernous'ko F.L., Ananievski I.M., Reshmin S. A. Control of Nonlinear Dynamical Systems. Methods and Applications. Comm. Control Engrg. Ser. Berlin, Heidelberg: Springer-Verlag, 2008. 396 p. DOI: 10.1007/978-3-540-70784-4
3. Chernousko F. L., Bolotnik N.N., Gradetsky V. G. Manipulation Robots: Dynamics, Control and Optimization. Boca Raton: CRC Press, 1994. 268 p.
4. Akulenko D.D., Bolotnik N. N., Kaplunov A. A. Some control modes of industrial manipulators. Izv. AN SSSR. Tekhnicheskaia Kibernetika, 1985. No. 6. P. 44-50.
5. Whittaker E. T. A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge: Cambridge University Press. 1988. 456 p. DOI: 10.1017/CB09780511608797
1 x
Figure 2. State plane (x, x').
¥
