NATIONAL NIVERSITY OF SINGAPORE FINAL EXAMINATION FOR THE DEGREE OF B.ENG ME 444 - DYNAMICS AND CONTROL OF ROBOTIC SYSTEMS October/November 994 - Time Allowed: 3 Hours INSTRCTIONS TO CANDIDATES:. This eamination paper contains SIX (6) questions and comprises TEN (0) pages.. Answer an FOR (4) questions. 3. All questions carr equal marks. 4. This is an open-book eamination. You ma open an number of books and notes.
Pg ME 444. a. Fig. shows a connecting (bent) rod (BJ) rigidl connecting the rotational joint J to the cuboid (ABCDEFGO). Let > Frame be fied and serve as the universe frame of reference. > The cuboid (ABCDEFGO), bent connecting rod (BJ), and joint J be all connected as one rigid assembl. 3> Frame H be attached rigidl to the rotational joint J and Frame M be fied rigidl onto the cuboid. That is, Frames H and M are attached rigidl to the cuboid -connecting-rod assembl bod. 4> Frame H be translating with respect to Frame. 5> Ais k be fied onto Frame H. 6> The cuboid-connecting-rod rigid assembl is rotating about ais k. At a certain instant of time, the origin of Frame H (joint J) is translating at a velocit of ( ) H T u = 3 m/sec, and the cuboid-connecting-rod assembl is rotating about ais k at an angular velocit of 0 rad/sec. At this same instant of time the assembl and frames are at the configuration indicated in Fig.. At that same instant of time, determine: i. The angular velocit of the cuboid-connecting-rod assembl with respect to Frame. ii. The translational velocit of point C (on the cuboid) with respect to Frame. (5 marks) D E 3 A O B C G F length of line segments: AD=CB= DE=CF= J OG=EF=3 h 30 deg OJ=0 k h Fig.
Pg 3 ME 444 b. The Intellede robot is a si-ais manipulator with all rotational joints. The first three ais intersect at a common point. Fig. is a schematic diagram of the Intellede robot showing the si rotational joint aes (indicated b the line segments protruding out of the links). We wish to solve the inverse kinematics of the position problem for this robot. Can the decoupling principle be used? If so, identif the decoupled subsstem b showing the equations relating the decoupled task subspace and the decoupled subset of joints. Indicate the number of degrees of freedom of this subsstem. If not, eplain wh? (0 marks) Fig.
Pg 4 ME 444. a. Frames M and C are attached rigidl to a cuboid as shown in Fig. 3. Frame is fied and serves as the universe frame of reference. The cube undergoes the following motion in the indicated sequence: > Rotation about the ais of Frame C b 30, then > Translation of (,, 3) along Frame C, then 3> Rotation about the ais of Frame M b 45, and then 4> Rotation about the ais of Frame b 60. Let T i and T i be the 4 4 homogeneous transformation matrices that C M describe the position and orientation of Frames C and M, respectivel, in after motion i. Find i. ii. iii. iv. v. TC TC TC 3 TC 4 TM 4 D E A 3 c B C c F G Fig. 3 line segment lengths: AD= DC=3 DE= (8 marks)
Pg 5 ME 444 b. Fig.4 shows a 4-ais robot with all rotational joints. The first joint rotates about a vertical ais while the net three joints rotate about horiontal aes parallel to the plane of Frame. Frame E is attached to the robot end-effector. A si ais force-torque sensor provides 3 force and 3 torque readings along and about the,, and aes of the sensor frame S. Frame S is coincident to the fied frame of reference. At the following stationar configuration: 05. 0 0 0 RE pe TE = = 0. 6 0. 436 0. 660 5, 0 0. 6 0. 789 0. 047 4 0 0 0 the robot end-effector is carring a cubic (0. m cube) paload of 0 kg. Determine the si readings of the si-ais force-torque sensor. Assume that the robot links are weightless. The gravitational force is pointing downward along the negative ais direction of Frame. The end-effector frame is at the base of the gripper. The center of mass of the paload is at its center and lies on the ais of the end-effector. The acceleration due to gravit is 9.8 m/s. (7 marks) s,u J 3 J 4 J J si-ais force sensor s,u s,u E paload center of mass E 0.05 m Fig. 4
Pg 6 ME 444 3. a. Frame C is attached rigidl to one corner of a cuboid as shown in Fig. 5. The ais of frame C is directed from point C to E. The ais of frame C is directed from point C to B. Determine the 4 4 homogeneous transformation matri TC that describes the position and orientation of frame C in frame. (5 marks) u u D E A 3 c C F c B G u Fig. 5 line segment lengths: AD= DC=3 DE= b. Fig. 6 shows a robot with two rotational joints J and J. Frame 0 is fied to ground. The first link (the rigid bent rod J ABJ which connectes joints J and J ) rotates about the -ais of frame 0. The bent rod J AB alwas lies on the plane of frame 0. Frame is attached to the end-effector as indicated in Fig. 6. The and aes of frame are normal and along the longitudinal ais, respectivel, of link. The robot is in motion, and at a certain time instant the following are known: > The robot is at the configuration shown in Fig. 6. > The ais of motion of joint J is parallel to the ais of frame 0. 3> The coordinates of J in frame 0 are (, 4, 6). 0 J 0 6 0 A 0 J q q B Fig. 6 link 4 30 deg line segments: link J A = 0 AB = 4 parallel to 0 B J = 6 parallel to 0 i. Assign a Cartesian coordinate frame to link (the bent rod) according to the Denavit-Hartenberg Convention. Sketch the robot with the coordinate frames on our eamination scripts.
Pg 7 ME 444 ii. Fill in the the following table of kinematic parameters: Link θ i r i d i α i iii. Draw the robot at the configuration q = 0 and q = 45. iv. Determine the position and orientation of the end-effector (frame ) in frame 0 as a function of the joint variables. Epress this position and orientation as a 4 4 homogeneous transformation matri 0 T. (0 marks) 4. A two degree-of-freedom manipulator shown in Fig. 7 moves an object of mass m in the vertical plane. The manipulator starts at rest from Position at time t = 0 (i.e. θ( 0 ) = θ, ρ( ) ρ, θ & 0 = ( 0 ) = 0, ρ&( 0 ) = 0), and moves to Position at time t = T seconds (i.e. θ( T ) = θ, ρ( ) ρ, θ & T = ( T ) = 0, ρ&( T ) = 0 ) as shown in Fig. 8. Y ρ(t) mass m θ(t) X Fig. 7 a. Derive the Lagrange dnamic equations of the manipulator b taking into account the object mass m and neglecting the masses of the manipulator links. ( marks) Y ρ Position θ 0 X θ ρ Position Fig. 8
Pg 8 ME 444 θ && ( t ) ρ&&( t ) a b t T T 0 0 T T t -a -b Fig. 9 b. Assume that the joint motions are performed with constant accelerations as shown in Fig. 9. For θ = 5 degrees, θ = 5 degrees, ρ = m, ρ = m, T = sec, determine the following: i. the magnitudes a and b of the joint accelerations θ && ( t ) and ρ&&( t ) respectivel. ii. the trajectories θ ( t ) and ρ( t ). iii. the actuator torque τ ( t ) and the actuator force f(t) required for this motion. 5. a. Consider the coupled nonlinear sstem && + 3 + = u + u, && + & cos + 3( ) = u 3(cos ) u, where u, u are the inputs and, are the outputs. i. What is the dimension of the state space of the sstem? (3 marks) ii. Choose state variables and write the sstem as a sstem of first order differential equations in state space. iii. Find an inverse dnamics control so that the closed loop sstem is linear and decoupled, with each subsstem having natural frequenc 0 radians and damping ratio 0.5. ( marks)
Pg 9 ME 444 b. Consider the nonlinear dnamic equation of a manipulator given b M( q )&& q + C( q, q& )& q + g( q ) = u, where M( q) denotes the inertia matri, C( q, q& )& q denotes the coriolis and centrifugal terms, g( q) denotes the gravitational terms, and u denotes the generalied inputs.se Lapunov analsis to show that the PD control law u = K q K q&, p d where K p, Kd are diagonal matrices of (positive) proportional and derivative gains respectivel, ields a stable nonlinear sstem. Give an epression for the stead state value of q. (3 marks) 6. a. Consider the simplified sstem shown in Fig. 0 representing an end effector contacting the environment along the direction labeled where f : input force eerted b the end effector f dist : a disturbance force m e : equivalent inertia of environment (positive) b e : damping constant (positive) k e : stiffness constant (positive). Fig. 0 The contact force which is given b f e = k e is required to track a constant desired force trajector f d. i. Choosing, & as state variables, and the force error as the output variable, i.e. = f f, d e epress the dnamic equations of the sstem in state space form.
Pg 0 ME 444 ii. If the force input is chosen as f = k k & + k f + f, 3 d dist write down all the conditions involving k, k, k 3, k e, be such that the stead state force error ss is ero when f d is a constant (i.e. a step input). ( marks) b. The Lagrange dnamic equations of a two degree of freedom cartesian manipulator is given b && = f f e, && = f f, e where, are the joint variables, f, f are the input forces, and f e, f e are the cartesian components of the contact force at the end effector. Consider the constraint surface shown in Fig. which is described as 08. + 0. 6. 4 = 0. The end effector is initiall in contact with the constraint surface and initial conditions are given b ( 0 ) = 3, ( 0 ) = 0, &( 0 ) = 0, &( 0 ) = 0. Y 4 constraint surface f n 0 3 X Fig. Assume that the constraint surface is frictionless and the contact force is alwas normal to the constraint surface. Find the input forces f, f which would maintain a constant normal contact force f n of magnitude N, and move the end effector of the manipulator along the constraint surface with a tangential velocit t m/s, where t is the time in seconds. (3 marks) END OF PAPER