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.a) Show by use of the contraction mapping theorem that if a -|w| > 0then the differential equation has a unique equilibrium.b) Assume that a = b = 1 and w =1/2.Use the contraction mappingtheorem together with a numerical algorithm to obtain an approxi-mated value of the unique equilibrium of the differential equation.4.Consider the function4 1 x1V (x1, x2) =[ x1 x2 ].-10 3 x2Is V (x1, x2) positive definite?5.Consider the functionV (x1, x2) =ax2 +2bx1x2 + cx2.1 2Show that if a >0 and ac > b2 then V (x1, x2) is positive definite.6.Consider the linear autonomous differential equation = Ax, x " IRn.Show that if there exists a pole of this equation at the origin of the complexplane, then the equation has an infinite number of equilibria.Hint: Here, the poles are the eigenvalues of A.7.An equilibrium xe " IRn is an isolated equilibrium of = f(x) if thereexists a real positive number > 0 such that there may not be anyequilibrium other than xe in &!, where&!={x " IRn : x - xe 0 that satisfies the abovethen the equilibrium xe is not isolated.Assume that xe is a non-isolated equilibrium.Answer true or false tothe following claims: 56 2 Mathematical Preliminariesa) the equilibrium xe may not be asymptotically stable;b) the equilibrium xe is stable.8.Consider the function f(x, y) : IR2 ! IR2,f1(x, y)f(x, y) =.f2(x, y)Assume that f(x, y) =0 !! x = 0 and y = 0.Does this imply thatf1(x, y) =0 !! x = 0 and y =0 ?9.Consider the following two differential equations: 1 = [x1 - ] +x2 - [x1 - ] [x1 - ]2 + x2 , x1(0) " IR2 2 = -[x1 - ] +x2 - x2 [x1 - ]2 + x2 , x2(0) " IR2where " IR is constant.Determine the equilibria.10.Consider the following second-order differential equation, +[y2 - 1] + y2 +1 =0, y(0), (0) " IR.Express this equation in the form = f(t, x).a) Is this equation linear in the state x ?b) What are the equilibrium points? Discuss.11.Consider the equation = f(x).Assume that xe = 0 " IRn is a stableequilibrium.Does this imply that the solutions x(t) are bounded for allt e" 0?12.Consider the equations 1 = x2 - x31 2 = -x1 - x32for which the origin is the unique equilibrium.Use the direct Lyapunov smethod (propose a Lyapunov function) to show that the origin is stable.13.Pick positive integer numbers m and n and appropriate constants a andb to make up a Lyapunov function of the formV (x1, x2) =ax2m + bx2n1 2in order to show stability of the origin fora) 1 = -2x32 2 =2x1 - x32 Problems 57b) 1 = -x3 + x31 2 2 = -x3 - x3.1 214.Theorem 2.4 allows us to conclude global uniform asymptotic stability ofan equilibrium of a differential equation.To show only uniform asymptoticstability (i.e.not global), the conditions of Theorem 2.4 that impose tothe Lyapunov function candidate V (t, x) to be:" (globally) positive definite;" radially unbounded;" (globally) decrescent and," for V (t, x), to be (globally) negative definite;must be replaced by:" locally positive definite;" locally decrescent and," for V (t, x), to be locally negative definite.If moreover, the differential equation is autonomous and the Lyapunovfunction candidate V (x) is independent of time, then the equilibrium is(locally) asymptotically stable provided that V (x) is locally positive def-inite and V (x) is locally negative definite.An application of the latter is illustrated next.Consider the model of a pendulum of length l and mass m concentratedat the end of the pendulum and subject to the action of gravity g, andwith viscous friction at the joint (let f >0 be the friction coefficient), i.e.ml2q + fq + mgl sin(q) =0, where q is the angular position with respect to the vertical.Rewrite themodel in the state-space form = f(x) with x =[q q]T.a) Determine the equilibria of this equation.b) Show asymptotic stability of the origin by use of the Lyapunov func-tion2"ml2 1 fV (q, q) =2mgl[1 - cos(q)] + q2 + " q + l mq. 2 2 l mc) Is V (q, q) a negative definite function?15.Complete the analysis of Example 2.15 by applying Lemma 2.2. 3Robot DynamicsRobot manipulators are articulated mechanical systems composed of linksconnected by joints.The joints are mainly of two types: revolute and prismatic.In this textbook we consider robot manipulators formed by an open kinematicchain as illustrated in Figure 3.1.q3z0 znz2 link 2joint njoint 2 z3link nq2qnz1# #link 1x1joint 1#x2#.x =# #.q1.xmy0x0Figure 3.1.Abstract diagram of an n-DOF robot manipulatorConsider the generic configuration of an articulated arm of n links shownin Figure 3.1.In order to derive the mathematical model of a robot one typ-ically starts by placing a 3-dimensional reference frame (e.g.in Cartesiancoordinates) at any location in the base of the robot.Here, axes will be la-beled with no distinction using {x y z} or {x0 y0 z0} or {x1 x2 x3}.Thelinks are numbered consecutively from the base (link 0) up to the end-effector(link n).The joints correspond to the contact points between the links and 60 3 Robot Dynamicsare numbered in such a way that the ith joint connects the ith to the (i- 1)thlink.Each joint is independently controlled through an actuator, which is usu-ally placed at that joint and the movement of the joints produces the relativemovement of the links.We temporarily denote by zi, the ith joint s axis ofmotion.The generalized joint coordinate denoted by qi, corresponds to theangular displacement around zi if the ith joint is revolute, or to the lineardisplacement along zi if the ith joint is prismatic.In the typical case wherethe actuators are placed at the joints among the links, the generalized jointcoordinates are named joint positions.Unless explicitly said otherwise, weassume that this is the case.z3z1z0q3z2z4q2q1q4# #x1#x2#x =x3y0x0Figure 3.2.Example of a 4-DOF robotExample 3.1.Figure 3.2 shows a 4-DOF manipulator.The placementof the axes zi as well as the joint coordinates, are illustrated in thisfigure.f&The joint positions corresponding to each joint of the robot, and whichare measured by sensors adequately placed at the actuators, that are usuallylocated at the joints themselves, are collected for analytical purposes, in thevector of joint positions q.Consequently, for a robot with n joints, that is,with n DOF (except for special cases, such as elastic-joints or flexible-linkrobots), the vector of joint positions q has n elements: 3 Robot Dynamics 61# #q1q2# ## #q =.# #.qni.e.q " IRn.On the other hand it is also of great interest, specifically from apractical viewpoint, to determine the position and orientation (posture) of therobot s end-effector since it is the latter that actually carries out the desiredtask.Such position and orientation are expressed with respect to the referenceframe placed at the base of the robot (e.g.a Cartesian frame {x0, y0, z0}) andeventually in terms of the so-called Euler angles.Such coordinates (and angles)are collected in the vector x of operational positions1#x #1x2# ## #x =.# #.xmwhere m d" n.In the case when the robot s end-effector can take any positionand orientation in the Euclidean space of dimension 3 (e.g.the room wherethe reader is at the moment), we have m = 6.On the other hand, if therobot s motion is on the plane (i.e.in dimension 2) and only the position ofthe end-effector is of interest, then m = 2.If, however, the orientation on theplane is of concern then, m =3.The direct kinematic model of a robot, describes the relation between thejoint position q and the position and orientation (posture) x of the robot send-effector.In other words, the direct kinematic model of a robot is a function : IRn ! IRm such thatx = (q)
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