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

Chupin, I. A.

Dolgii, Y. F.

Sesekin, A. N.

Ural mathematical journal

Dolgii, Yurii F.

Sesekin, Alexander N.

Chupin, Ilya A.

Ural Mathematical Journal (UMJ)

IMPULSE CONTROL OF THE MANIPULATION ROBOT

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

URAL MATHEMATICAL JOURNAL, Vol. 5, No. 2, 2019, pp. 13–20DOI: 10.15826/umj.2019.2.002IMPULSE CONTROL OF THE MANIPULATION ROBOTYurii F. Dolgii1,2,†, Alexander N. Sesekin1,2,††, Ilya A. Chupin1,†††1Ural Federal University,19 Mira str., Ekaterinburg, 620002, Russia2Krasovskii Institute of Mathematics and Mechanics,Ural Branch of the Russian Academy of Sciences,16 S. Kovalevskaya Str., Ekaterinburg, 620990, Russia†juri.dolgy@urfu.ru, ††a.n.sesekin@urfu.ru, †††mr.tchupin@yandex.ruAbstract: A nonlinear control problem for a manipulation robot is considered. The solvability conditions forthe problem are obtained in the class of special impulse controls. To achieve the control goal, the kinetic energyof the manipulation robot is used. When finding analytical formulas for controls, the classical first integralsof Lagrangian mechanics were used. The effectiveness of the proposed algorithm is illustrated by computersimulation.Keywords: Manipulation robot, Impulse controls, First integrals.IntroductionThe purpose of controlling the manipulation robot is to transfer it from the initial positionto the final. The significant nonlinearity of the mathematical model describing the movements ofthe manipulation robot does not allow the use of the methods of the mathematical control theorydirectly for the original model. Decomposition methods make it possible to reduce the dimension ofthe 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 ofa 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 replacedwith an integrable controlled model. The Pontryagin maximum principle was used to find controlsin the performance problem. The use of special impulse controls in this work allows the kineticenergy of the manipulation robot to be used to reach the final position. This approach preservesthe 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 ofnumerical simulation of the algorithm.1. Mathematical statement of the problemA 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 1indicates the base of the robotic arm, 2 is a rack of vertically oriented shaft. This shaft is rigidlyconnected to the guide beam 4 and hand 5.Kinetic energy of the manipulation robot is determined by the formulaT =12(m1z′2 + (J1 + J2)ϕ′2 +m2(z′2 + x′2 + x2ϕ′2)),14 Yurii F. Dolgii, Alexander N. Sesekin, Ilya A. ChupinX'ZXZ'12345Y'0'0YxFigure 1where x is the coordinate of the center of mass of the arm, x > 0; z is the height of the arm, z > 0;ϕ is the angle of rotation of the hand; m1 is the total mass of the shaft that rotates the arm, and theguides that implement the horizontal displacement of the arm; m2 is the mass of the arm; J1 is themoment of inertia of the shaft and guides relative to the vertical axis; J2 is the central moment ofinertia 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 theform [3, p. 263]z′′ + g = u1, (1.1)x′′ − xϕ′2 = u2, (1.2)ϕ′′ + α2(x2ϕ′)′= u3. (1.3)Here u1 = P/(m1 +m2), u2 = F/m2, u3 = M/(J1 + J2) are the control force actions, where P isthe magnitude of the longitudinal force acting along the vertical axis, F is the magnitude of thelongitudinal force acting along the horizontal guides, M is the magnitude of the moment of forcedirected along the vertical axis, α =√m2/(J1 + J2).It is required to find the controls u1, u2, u3 that translate the system (1.1)–(1.3) from the initialequilibrium position (z0, x0, ϕ0)⊤ , z0 > 0, x0 > 0, to the given end position (z∗, x∗, ϕ∗)⊤, z∗ > 0,x∗ > 0, z∗ 6= z0, x∗ 6= x0, ϕ∗ 6= ϕ0.To solve this problem, we use a special set U of impulse controls defined by the formulasu1(t) = ż0δ(t), u2(t) = ẋ0δ(t), u3(t) = ϕ̇0δ(t), t ∈ R,where δ(·) is the Dirac impulse function. Impulse controls at the initial moment of time t = 0 tothe dynamic system (1.1)–(1.3), which are in equilibrium, report the initial velocities z′(0) = ż0,x′(0) = ẋ0, ϕ′(0) = ϕ̇0. The initial speeds are control parameters, the choice of which should ensurethat the dynamic system (1.1)–(1.3) falls into the final position.2. Impulse control parametersThe problem is divided into two problems. In the first problem, for the equation (1.1) thatdetermines the vertical movement of the manipulation robot, we use the set U1 of impulse controlsdefined by the formula u1(t) = ż0δ(t), t ∈ R. It is required to find the control u1, which transfersthe dynamical system (1.1) from the initial equilibrium position z0, z0 > 0, to the given finalImpulse control of manipulation robot 15position z∗, z∗ > 0, z∗ 6= 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 impulsecontrols defined by the formulas u2(t) = ẋ0δ(t), u3(t) = ϕ̇0δ(t), t ∈ R. It is required to findcontrols u2, u3 that translate the dynamical system (1.2), (1.3) from the initial equilibrium position(x0, ϕ0)⊤ , x0 > 0, to the given end position (x∗, ϕ∗)⊤ , x∗ > 0, x∗ 6= x0, ϕ∗ 6= ϕ0.Lemma 1. In the set U1 there are impulse controls u1 that transfer the dynamical system (1.1)from an arbitrary initial position z0, z0 > 0, to an arbitrary end position z∗, z∗ > 0, z∗ 6= z0.P r o o f. The impulse control u1 ∈ U1 provides the dynamic system (1.1) at the initial momentt = 0 the initial velocity z′(0) = ż0. For t > 0, the motion of a free dynamic system is determinedby the differential equation z′′+ g = 0 with the initial conditions z(0) = z0, z′(0) = ż0. The verticalmovement is described by the formula z(t) = −gt2/2 + ż0t + z0, t > 0. For any τ1 > 0 there is aunique value of the control parameter z0 for which the equality z(τ1) = z∗, ż0 = τ−11 (z∗ − z0)+gτ1/2is true.We show that the velocity at the finite moment of time is minimal in absolute value ifτ1 =√2|z∗ − z0|/g. The velocity at the finite moment of time is determined by the formulaz′(τ1) = −gτ1 + ż0 = −gτ1/2 + τ−11 (z∗ − z0) .If z∗ > z0 > 0, then we have z′(τ1) = 0 under the condition τ1 =√2(z∗ − z0)/g. The func-tion f(τ) = g1/2 + τ−1 (z0 − z∗) has a minimum at τ = τ1 =√2 (z0 − z∗) /g which is equal to√2g (z0 − z∗) if z0 > z∗ > 0.The impulse controls u2, u3 ∈ U2 prescribe the dynamic system (1.2), (1.3) at the initial momentt = 0 the initial speeds x′(0) = ẋ0, ϕ′(0) = ϕ̇0. For t > 0 the motion of a free dynamic systemis determined by differential equations x′′ − xϕ′2 = 0, ϕ′′ + α2(x2ϕ′)′= 0 with initial conditionsx(0) = x0, ϕ(0) = ϕ0, x′(0) = ẋ0, ϕ′(0) = ϕ̇0. For the horizontal free movement of the manipulationrobot, kinetic energy and momentum are kept the same [5]T2 =12((J1 + J2)ϕ′2 +m2(x′2 + x2ϕ′2))= const,p2 =∂T2∂ϕ′= (J1 + J2 +m2x2)ϕ′ = const.When describing horizontal motion for t > 0, we replace the system (1.1), (1.2) with the system ofdifferential equationsα2x′2 + (1 + α2x2)ϕ′2 = c1, (2.1)(1 + α2x2)ϕ′ = c2, (2.2)wherec1 = α2ẋ20 + (1 + α2x20)ϕ̇20, c2 = (1 + α2x20)ϕ̇0.Lemma 2. Let the conditions|ϕ∗ − ϕ0| ≤x∗∫x0√1 + α2x20ds√(1 + α2s2)(s2 − x20), 0 < x0 < x∗, (2.3)|ϕ∗ − ϕ0| ≤x0∫x∗√1 + α2x2∗ds√(1 + α2s2)(s2 − x2∗), 0 < x∗ < x0, (2.4)16 Yurii F. Dolgii, Alexander N. Sesekin, Ilya A. Chupinhold. Then in the set U2 there are impulse controls u1, u2 that move the dynamical system (1.2),(1.3) from the starting position (x0, ϕ0)⊤ , x0 > 0, to the ending position (x∗, ϕ∗)⊤ , x∗ > 0, x∗ 6= x0,ϕ∗ 6= ϕ0.P r o o f. Under the condition ẋ0 6= 0, the system of differential equations (2.1), (2.2) istransformed to the following formx′ = sgn ẋ0√ẋ20 + (1 + α2x20) ϕ̇20x2 − x201 + α2x2, x ∈ X, (2.5)ϕ′ =(1 + α2x20)ϕ̇01 + α2x2, x ∈ X, (2.6)whereX ={x ∈ R+ : ẋ20 + (1 + α2x20)ϕ̇20x2 − x201 + α2x2≥ 0}.To move the motion of the dynamical system (2.5), (2.6) from the initial to the final position,the control parameters ẋ0, ϕ̇0 must satisfy the conditionsϕ̇0 6= 0, sgn ẋ0 = sgn (x∗ − x0), sgn ϕ̇0 = sgn (ϕ∗ − ϕ0).We introduce the parameter p = |ẋ0|/|ϕ̇0|. Now the description of the set X = X(p) is simplified.As a result, we haveX(p) = R+ under p ≥ x0√1 + α2x20,X(p) = R+/(0, x1(p)) under p < x0√1 + α2x20,wherex1(p) =√x20(1 + α2x20)− p21 + α2x20+ α2p2.The equation (2.5) is converted tox′ = sgn (x∗ − x0)|ϕ̇0|√p2 + (1 + α2x20)x2 − x201 + α2x2, x ∈ X. (2.7)It is also valid for p = 0. Using (2.7) and (2.6), we obtain a differential equation for finding thetrajectory of a horizontal movementdxdϕ=sgn (x− x0)(1 + α2x2)sgn (ϕ∗ − ϕ0)(1 + α2x20)√p2 + (1 + α2x20)x2 − x201 + α2x2, x ∈ X(p).Integrating the differential equation, we find the equation of the trajectory of a horizontal movementsgn (x∗ − x0)x∫x0(1 + α2x20)ds(1 + α2s2)√p2 + (1 + α2x20)s2 − x201 + α2s2= sgn (ϕ∗ − ϕ0)(ϕ− ϕ0), x ∈ X(p).The trajectory passes through the end point if the condition is true∣∣∣∣∣x∗∫x0(1 + α2x20)ds(1 + α2s2)√p2 + (1 + α2x20)s2 − x201 + α2s2∣∣∣∣∣= |ϕ∗ − ϕ0|, x∗ ∈ X(p).Impulse control of manipulation robot 17We select the value of the parameter p to satisfy the condition obtained. For 0 < x0 < x∗, therequired value of the parameter p is determined by the equationx∗∫x0ds(1 + α2s2)√p2 + (1 + α2x20)s2 − x201 + α2s2=|ϕ∗ − ϕ0|1 + α2x20, 0 ≤ p < +∞. (2.8)For 0 < x∗ < x0, the required value of the parameter p is determined by the equationx0∫x∗ds(1 + α2s2)√p2 + (1 + α2x20)s2 − x201 + α2s2=|ϕ∗ − ϕ0|1 + α2x20, pkp ≤ p < +∞, (2.9)wherepkp =√(1 + α2x20)(x20 − x2∗)1 + α2x2∗.The equation (2.8) has a unique solution p = p∗ under the condition (2.3) and equation (2.9) alsohas 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) = ẋ0δ(t), u3(t) = ϕ̇0δ(t) (t ∈ R) bring the dynamicsystem (1.2), (1.3) to the given final position, if the control parameters are determined by theformulasẋ0 = |ẋ0| sgn (x∗ − x0), ϕ̇0 = |ϕ̇0| sgn (ϕ∗ − ϕ0), |ẋ0| = p∗|ϕ̇0|,where p∗ is the root of the equation (2.8) for 0 < x0 < x∗, and for 0 < x∗ < x0 there is a root ofthe equation (2.9).Integrating the differential equation (2.7), we find the arrival time of the motion of the dynamicalsystem (1.2), (1.3) at the end pointτ2 =1|ϕ̇0|∣∣∣∣∣x∗∫x0ds√p2∗ + (1 + α2x20)s2 − x201 + α2s2∣∣∣∣∣.We synchronize the arrival times of the movements of the dynamical systems (1.1)–(1.3) to theend points. Theorem 1. Let the conditions of Lemma 2 be satisfied. Then the values of the parame-ters of the impulse controls that move the dynamical system (1.1)–(1.3) from the initial position(z0, x0, ϕ0)⊤ , z0, x0 > 0, to end position (z∗, x∗, ϕ∗)⊤ , z∗ > 0, x∗ > 0, z∗ 6= z0, x∗ 6= x0, ϕ∗ 6= ϕ0are defined by formulasż0 =√2g(z∗ − z0) for 0 < z0 < z∗,ż0 = 0 for 0 < z∗ < z0,ϕ̇0 =√g sgn (ϕ∗ − ϕ0)√2|z∗ − z0|∣∣∣∣∣x∗∫x0ds√p2∗ + (1 + α2x20)s2 − x201 + α2s2∣∣∣∣∣,ẋ0 =√gp∗√2|z∗ − z0|x∗∫x0ds√p2∗ + (1 + α2x20)s2 − x201 + α2s2.18 Yurii F. Dolgii, Alexander N. Sesekin, Ilya A. ChupinP r o o f. Using Lemma 1 and the arrival time τ1 of the motion of the dynamical system (1.1) tothe end point, we find the value of the control parameter ż0. The synchronization condition τ1 = τ2of the arrival times of the motions of dynamical systems (1.1)–(1.3) at end points determinesthe arrival time τ =√2|z∗ − z0|/g of dynamic system movements (1.1)–(1.3) to the end point(z∗, x∗, ϕ∗)⊤ and the equation for the control parameter ϕ̇0. From this equation we find|ϕ̇0| =√g√2|z∗ − z0|∣∣∣∣∣x∗∫x0ds√p2∗ + (1 + α2x20)s2 − x201 + α2s2∣∣∣∣∣.Using Lemma 2, we find the control parameters ϕ̇0, ẋ0. 3. Stabilization of manipulation robot in a final positionWhen stabilizing the manipulation robot in a small neighborhood of the final position, we usespecial positional controls, the choice of which turns the final position into a stable equilibriumposition of the controlled system. For this purpose we use substitutions for coordinatesz = z∗ + ẑ, x = x∗ + x̂, ϕ = ϕ∗ + ϕ̂and controlsu1 = û1 + g, u2 = û2, u3 = û3(1 + α2x2∗).In a small neighborhood of the final equilibrium, the controlled system (1.1)–(1.3) is replaced bythe following controlled systemẑ′′ = û1, x̂′′ = û2, ϕ̂′′ = û3. (3.1)We find the stabilizing control using the theory of optimal stabilization for linear systems withquadratic quality criteria. Choosing the quality criterionJ1 =+∞∫0(ẑ2(t) + k21 ẑ′2(t) + û21(t))dt, k1 > 0, (3.2)for the first control of the system (3.1), we find the stabilizing controlû1 = −ẑ −√k21 + 2ẑ′.We also can find stabilizing controls for the second and third equations in (3.1)û2 = −x̂−√k22 + 2x̂′, û3 = −ϕ̂−√k23 + 2ϕ̂′,using a quality criteria similar to (3.2) with constants k2 and k2, respectively.4. Numerical modelingIn the numerical simulation of the system motions (1.1)–(1.3), the following values of theparameters of the mechanical system were usedm1 = 20, m2 = 8, J1 = 12, J2 = 6, g = 9.8.Impulse control of manipulation robot 19The start and the end positions are z0 = 0, x0 = 0, ϕ0 = 0 and z∗ = 1.4, x∗ = 0.5, ϕ∗ = 1.2.We take the controlsu1(t, z) = upr1 (t) + ups1 (z), u2(t, x) = upr2 (t) + ups2 (x), u3(t, ϕ) = upr3 (t) + ups3 (ϕ).The program control is defined by formulasupr1 (t) = ż0δ(t), upr2 (t) = ẋ0δ(t), upr3 (t) = ϕ̇0δ(t), t ∈ R,where the parameters are given by formulasż0 =√2gz∗, ẋ0 = p∗ϕ̇0, ϕ̇0 =√g2z∗x∗∫0√1 + α2s2p2∗(1 + α2s2) + s2ds.Here p = p∗ is the positive root of the equationx∗∫0ds√(1 + α2s2)((1 + α2s2)p2 + s2)= ϕ∗.Impulse controls moves the mechanical system into equilibrium, the initial speeds arez′(+0) = ż0, x′(+0) = ẋ0, ϕ̇(+0) = ϕ̇0.We also consider software controls in the form of rectangular impulses, which are approximationsof ideal impulsesupr1 (t) = ż0δ∆(t), upr2 (t) = ẋ0δ∆(t), upr3 (t) = ϕ̇0δ∆(t), t ∈ R,whereδ∆(t) = 1/∆, t ∈ (0,∆), δ∆(t) = 0, t ∈ R/(0,∆), ∆ = 0.1.For these controls, the initial velocities of the equilibrium mechanical system are determined bythe formulas z′(0) = 0, x′(0) = 0, ϕ̇(0) = 0.Positional controls are determined by the following formulasups1 (z) = 0, 0 < z ≤ z∗ − ǫ1,ups1 (z) = g − (z − z∗)−√k21 + 2 z′, z > z∗ − ǫ1,ups2 (x) = 0, 0 < x ≤ x∗ − ǫ2,ups2 (x) = −(x− x∗)−√k22 + 2x′, x > x∗ − ǫ2,ups3 (ϕ) = 0, 0 < ϕ ≤ ϕ∗ − ǫ3,ups3 (ϕ) = −(1 + α2x2∗)((ϕ− ϕ∗) +√k23 + 2ϕ′), ϕ > ϕ∗ − ǫ3,wherek1 = 1, k2 = 1, k3 = 1, ǫ1 = 0.1, ǫ2 = 0.1, ǫ3 = 0.1.In the final position, the following conditions must be met|z(τf )| < ǫ, |x(τf )| < ǫ, |ϕ(τf )| < ǫ, |z′(τf )| < ǫ, |x′(τf )| < ǫ, |ϕ′(τf )| < ǫ.In the computational experiment, we assumed that ǫ = 0.01.The time movement to the final position is τf = 7.5 sec. for impulse controls and τf = 7.51 sec.for rectangular impulses approximating ideal impulse actions. Projections of phase trajectories onstate planes (x, x′) and (ϕ,ϕ′) are shown in Fig. 2 and Fig. 3.The trajectories corresponding to impulse controls are shown in blue, the trajectories corre-sponding to the approximations of ideal impulse controls are shown in brown.20 Yurii F. Dolgii, Alexander N. Sesekin, Ilya A. Chupin0.2 0.4 0.6 0.8 1.0 1.2x-0.20.20.40.60.81.01.2x′Figure 2. State plane (x, x′).0.5 1.0 1.5φ0.51.01.52.0φ′Figure 3. State plane (ϕ, ϕ′).5. ConclusionThe impulse control is constructed in the work that transfers the manipulator from a givenposition to its final position. A computational experiment showing the efficiency of the proposedalgorithm is presented. The proposed algorithm is simulated in the case when the ideal impulse isapproximated by the usual bounded control.REFERENCES1. Kozowski K. Modelling and Identification in Robotics. Ser. Adv. Ind. Control. London: Springer–Verlag,1998. 261 p. DOI: 10.1007/978-1-4471-0429-22. Chernous’ko F. L., Ananievski I.M., Reshmin S.A. Control of Nonlinear Dynamical Systems. Meth-ods and Applications. Comm. Control Engrg. Ser. Berlin, Heidelberg: Springer–Verlag, 2008. 396 p.DOI: 10.1007/978-3-540-70784-43. Chernousko F. L., Bolotnik N.N., Gradetsky V.G. Manipulation Robots: Dynamics, Control and Opti-mization. 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/CBO9780511608797

Impulse Control of the Manipulation Robot

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. © 2019, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of Russian Academy of Sciences. All rights reserved

Impulse control of the manipulation robot

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Impulse Control of the Manipulation Robot

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Ural Mathematical Journal (UMJ)

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin