MDP646: ROBOTICS ENGINEERING Mechanical Design & Production Department Faculty of Engineering Cairo University Egypt Prof. Said M. Megahed
APPENDIX A: PROBLEM SETS AND PROJECTS Problem Set # Due 3 rd week P. The basic components of a robot are: mechanical structure, tooling, drives, sensors, and controller. Make an Internet Search to briefly describe the main function(s) of each of these components. P. Make an Internet Search about: a- Industrial robots b- Mobile Robots c- Walking Machines d- Remotely Operated Vehicles (ROV) e- Parallel Robot f- Robot Tooling and Grippers P3. With the aid of Internet search write the specifications of the industrial robots shown in Figure below. Also, suggest a specification chart for selecting a robot of type RRRRRR to perform spot and arc welding, and another for painting operations. P4. Sketch three different types of a robot end effector used for: pick-and place, loading a CNC machine, and inspection applications. P5. Drives are one of the basic elements of a robot arm. What are the characteristics of a good motor for a robot? Make a complete comparison between the advantages and disadvantages of all types of robot drives. Using Internet search, write the standard specification of each drive type (electrical, pneumatic, and hydraulic). P6. The measurement of the linear/angular displacement, linear/angular velocity, and the force/ torque of a robot prismatic/revolute joint are important for robot control. Select three of these parameters and with the aid of Internet Search sketch the corresponding
sensing device for measuring them explaining their principle of operation. P7. Explain with aid of sketches the following robotic terms: a- Robot Manipulator, Arm, hand, wrist, and gripper giving the function of each b- Degree of mobility, Degree of freedom, and Degree of flexibility of a robot arm giving examples whenever necessary. c- Versatility, Auto-adaptability, Robot reach, Compliance, Payload, and Tool velocity. d- Repeatability, Accuracy, Control Resolution, and Spatial Resolution. P8. Prove that repeatability is always less than accuracy. Problem Set # Due 5 th week P. A certain DC servomotor used to actuate a robot joint has a torque constant of N.m/A, and a voltage constant of 5V/Kr/min. The armature resistance is.5 Ω. At a particular moment during the robot cycle, the joint is not moving and a voltage of 3 V is applied to the motor. a- Determine the torque of the motor immediately after the voltage is applied. b- Determine the back-emf and the corresponding torque at 5 and 75 rpm. c- If there is no resistance torques and no inductance of the armature windings operating to reduce the speed of the motor, determine the maximum theoretical speed of the motor when the input voltage is 3 V. d- If the resisting torques due to friction and the payload being carried by the robot are 6 N.m, determine the maximum theoretical speed of the motor when the input voltage is 3 V. P. Describe with the aid of sketches an optical encoder to measure the displacement and velocity of a rotating shaft and determine the resolution of an optical encoder of six tracks? Nine tracks? Twelve tracks? And a stepper motor with a step angle of 5? P3. A certain potentiometer is to be used as the feedback device to indicate position of the output link of a rotational robot joint. The excitation voltage of the potentiometer equals V, and the total wiper travel of the potentiometer is 4, the wiper arm is directly connected to the rotational joint so that a given rotation of the joint corresponds to an equal rotation of the wiper arm. Determine the voltage constant of the potentiometer k p. If the robot joint is actuated to a certain angle, causing the wiper position to be 5, determine the resulting output voltage of the potentiometer. P4. A two axis Cartesian robot has a reach of 6 mm in the z direction and a reach of 75 mm in the y direction. The robot controller is sending 8-bit D/A and A/D signals (DACS) for control and feedback. Determine the overall resolution and accuracy of this robot. Assume that the gripper is capable of finger motion parallel to the y direction with a range of 75 mm. If the same controller controls the gripper motor, determine the worst 3
accuracy in a pick and place operation. APPENDIX A: Problem Sets & Projects P5. A xyz rectangular Cartesian robotic manipulator has reach spans of 4, 38, and 4 mm in the x, y, and z directions respectively. The robot controller is sending bit D/A (DACS) command signals for control and feedback. Determine the worst spatial resolution and accuracy of this robot in assembly processes if the mechanical inaccuracies are. mm. P6. One of the axes of an RRP robot is a sliding mechanism with a total travel of.75 m. The robot control memory has a -bit capacity. In addition, it has been observed that the mechanical inaccuracies associated with moving the arm to any given programmed point form a normally distributed random variable with the mean at the taught point and the standard deviation equal to.5 mm. Assume that the total standard deviation is isotropic (it is equal in all directions). By definition, threestandard deviations include all of the mechanical errors in the arm movement. With these definitions and assumptions, determine the following: the control and spatial resolution for this axis, the defined control and spatial accuracy of the robot for this axis, and the repeatability of the robot Problem Set # 3 Due 7 th week P. A two-link robot arm of type RR is working in horizontal plane and has its links of lengths (l = 5 mm and l = 3 mm). a- Sketch this robot in a general position of its joint angles q and q b- Draw to scale its working area for each of the following ranges of motion of its joints: c- o q o and o q 8 o & ii- o q 5 o and o q 9 o d- Determine the robot DKPM and IKPM. e- Calculate the robot arm Jacobean and Hessian matrices as function of q and q where q and q are the joints angles. f- Find the expressions of the applied torques at its joints. The links masses are m and m. These masses are concentrated at the links mid points C and C. P. Repeat Problem if the robot works in vertical plane. P3 A two link planar robot arm of type RP has its links of lengths (l = 45 mm and l = 5 mm). The links' masses, and inertias about mass centers (links' mid lengths) are respectively: m, I & m, I. The robot is working in a vertical plan plane to resist a certain vertical load P (N) at its end effector TCP. a- Sketch this robot in a general position of its joint angles q and q b- Draw to scale its working area for each of the following ranges of motion of its joints: (-9 o q 5 o and o q 3 mm) where q and q are the joints angles c- Determine the robot DKPM and IKPM. d- Determine the Jacobean and Hessian matrices of this robot. 4
e- Determine the required joints' torques τ and τ as function of (l, m, I, l, m, I, q, q, q, q, q, q & P). P4 A three link planar robot arm of type RRR has its links of lengths (l = 6 mm, l = 4 mm, and l 3 =3 mm). The links' masses and inertias about mass centers (links' mid lengths) are respectively: (m, I & m, I & m 3, I 3 ). The robot is working in a horizontal plan plane to resist a certain vertical load P (N) at its end effector TCP. a- Draw to scale its working area for each of the following ranges of motion of its joints: (-9 o q 8 o, -9 o q 5 o and -9 o q 3 9 o ) where q, q and q 3 are the joints angles. b- Determine the robot DKPM and IKPM. c- Determine the Jacobean and Hessian matrices of this robot. d- Determine the required joints' torques τ, τ and τ 3 as function of the applied load P, robot geometrical and dynamical parameters, joints angles and their time derivatives. P4. Study the robot arms shown in figure and draw the working space for each assuming the necessary missing data. SCARA Robot 5
Problem Set # 4 Due 9 th week P. Find the 3x3 matrices, which describe the following motions in, -D: a. A 6 rotation about the origin. b. A 6 rotation about the point x=, y=. c. A translation of units in the x-direction followed by a 9 rotation about the origin. P. A -D rigid motion takes the points (,) and (,) to (.73,.73) and (.68,.5) respectively. Find the 3x3 matrix, which effects this transformation. P3. If R xyz = (O, X, Y, Z) is a fixed Coordinate frame of reference and R uvw = (O, U, V, W) is a rotating frame, what is the 3x3 rotation matrix R in the following cases: a. a rotation of 3º about the OZ axis, followed by a rotation of 6º about the OX axis, followed by a rotation of 9º about the OY axis? b. a rotation of an angle φ about the OX axis, followed by a rotation of an angle ψ about the OW axis, followed by a rotation of an angle θ about the OY axis?.find another sequence of rotations that is different, which results the same rotation matrix. c. a rotation of an angle α about the OX axis, followed by a translation of b unit of distance along the OZ axis, followed by a rotation of angle φ about the OV axis? P4. For the figures shown below, find the 4x4 homogeneous transformation matrices T i, i + and T,i for i=,, 3, 4, 5 for Fig. a and for i=,,3,4 for Fig. b. (Note: can you find the answer by observation based on the geometric interpretation of homogeneous transformation matrix?) c z 3 b z 5 x 5 z 3 y 3 x 3 d x 4 x 3 y 3 5 in y 5 z 4 e z 4 y 4 y in a x 4 y 4 z 4 in y x z x z x z x y z y x Fig. a y z y x 3 in Fig. b 3.5 in P5. Find the 4x4 matrices corresponding to the following 3-D rigid transformations: a. A rotation of 9 about the x-axis, followed by a translation of units in the 6
z-direction. b. A translation of units in the z-direction followed by a 9 rotation about the x-axis. c. A translation of units in the z-direction followed by a 9 rotation about the x-axis, followed by a translation of - units in the z-direction. P6. Find the rigid body transformations which take the points (,,), (,,) and (,,) respectively to: (a) (,,), (3,,) and (,,), (b). (,,), (,,) and (-,,), and (c). (,,), (,,) and (-,,) P7. Describe the particular robot system, using sketches when necessary, to illustrate a robot of the following types: PPR, RPR, PRR, PPP, and RRR. Give the name of the robot configuration in each case. P8. Define with the aid of neat sketches Denavit-Hartenberg (DH) Parameters for the i th link of a robot arm and determine with proof an analytical expression for its elementary HTM (T i, i+ ). Find also the inverse of this matrix (T i, i+ ) -. P9. The robot arm shown in Figure is of SCARA type (RRRP). Sketch the robot equivalent kinematical diagram and determine the robot arm elementary HTM s (T i, i+ ) for i = to 5 according to Denavit-Hartenberg notation, its generalized HTM T 5 and hence define its DKPM and IKPM (assume any messing data). Verify your results using vector analysis P. A cube of -meter edge length is positioned such that its corners have the following coordinates: [(,,); (,,); (,,); (,,); (,,); (,,), (,,); (,,)]. The position of the cube corners, (in meters), have been indicated relative to the reference coordinate system (xyz). From its current position, the cube is rotated 9 about the y-axis, followed by a 9 about the z-axis, followed by translation of + meters in the x direction. a- Assuming a coordinate system (uvw) fixed to the cube and initially coincident with the reference coordinate system (xyz), sketch the required transformation showing the moving coordinate system (u, v, w) with respect to the reference coordinate system (x, y, z) after each rotation or translation. b- Find the resulting HTM (T) after the move which satisfies the following relation: [x y z ] t = T [u v w ] t c- Find the new coordinates of the cube corners in the xyz frame after the move. d- Find the inverse of the HTM found in (b) T -. 7
Problem set # 5 Due 3 th week P. Given the Stanford arm as following figure, where d=. a- Find the link parameters for the arm. (Note: d3 is a prismatic joint variable, other joints are rotational joints; the link coordinate frames have been established as shown in the Fig.). b- Find the forward kinematic model for the arm and represent it in homogeneous matrix form. (Suggestion: the matrix chain product can be done either by hand or using Matlab symbolic toolbox if you have the software, simplify the results) c- Solving the above forward kinematics (i.e., represent the orientation with Yaw-Pitch-Roll angels). P. A two degree-of-freedom manipulator is shown in figure. The length of each link is m. a- establish its link coordinate frames and find the elementary HTM s T, T and the generalized HTM. b- Find the inverse kinematics solution for this manipulator (i.e., assuming you know the position/orientation of the robot tip, solve for the joint variables). c- Find the Jacobean matrix in the base coordinate frame for the two degree-of-freedom manipulator as shown in below (the length of each link is l; mass of the links are m and m). 8
P3 The six-link robot arm shown in Figure is of type RRRRRR. The robot axes are either parallel or perpendicular (Assume any missing data). a- Draw its equivalent kinematics diagram and establish the necessary orthonormal link coordinate systems (R i = o i, x i, y i, z i ) indicating those of the base R and tool R 7 according to DH notation. b- Define the necessary DH parameters specifying the type of the generalized coordinate (q i ) of each joint and write the corresponding elementary HTM's (T i, i+, i =,,, 6) in symbolic form. c- If the elements of the generalized HTM (T 7 ) of this robot are known, define its DKPM using Euler angles (Ψabout z-axis, followed by θ about x-axis, followed by ϕ about z-axis) for tool orientation and Cartesian coordinates for the position of its center (TCP). d- Find the Robot Jacobean Matrix. e- Find the robot IKPM P4. Establish the orthonormal link coordinate frames Ri = ( ox i, i, yz i, i) for i=,,, 6 for the PUMA 6 robot arm shown in the figure and complete the table below. PUMA 6 robot arm link coordinate parameters Joint i 3 4 5 6 θ i α i a i r i 9
P5. The robot arm shown in Figure is of type RRRRRR and its axes are either parallel or perpendicular. a- Determine the robot elementary HTM's (T i, i+ ) for i = to 6 according to DH symbolic notation if a coordinate system R i+ = ( o i+, x i+, y i+, z i+ ) is attached to link i. b- If the elements of the generalized HTM (T 7 ) of this robot are known, define its DKPM using Roll φ about z-axis, Pitch φ about y-axis, and Yaw φ 3 about x-axis for tool orientation and Spherical Coordinates (r, θ, and φ) for the position of its center. c- Find the Robot Jacobean Matrix. d- Find the robot IKPM P6. Find the direct and inverse kinematics relations for the robot arm shown in Figure (assume any missing data). P7. The robot arm shown in the Figure of P3 is of type RRRRR. Find the elements of its Jacobean matrix if the position of its end effector is defined by the Cartesian coordinates (l e, m e, n e ) and its orientation is defined by Euler parameters (p e, q e, r e, s e ). P8. The expressions of the non-zero elements of the HTM T 7 of a six-link robot arm are: t = (C*C4*C5+S*S5)*C6+C*S4*S6 t = C*C4*S5-S*C5 t 3 = (C*C4*C5+S*S5)*S6-C*S4*C6 t 4 = (C*C4*S5-S*C5)*r6+S*q3+C*a t = (S*C4*C5-C*S5)*C6+S*S4*S6 t = S*C4*S5+C*C5 t 3 = (S*C4*C5-C*S5)*S6-S*S4*C6 t 4 = (S*C4*S5+C*C5)*r6-C*q3+S*a t 3 = S4*C5*C6-C4*S6 t 3 = S4*S5 t 33 = S4*C5*S6+C4*C6 t 34 = S4*S5*r6+q If C i = Cos q i and S i = Sin q i define the robot arm type and determine its direct and inverse kinematics position models (DKPM &IKPM).
Problem set # 6 Due 5 th week P: A two degree-of-freedom manipulator is shown in the figure below. Given that the length of each link is l; mass of the links are m and m; derive the dynamic model of this two-link robot arm using Lagrange-Euler method. Assume the mass is equally distributed and all the products of inertia are zero, with the pseudo-inertia matrices are: = 3 m m l m l m l J = 3 m l m l m l m J Z x y x y Z m m θ θ L L
Lab Work and Course Project Modeling, Simulation, Animation and Experimentation of FANUC Robot Arm Equipments FANUC Robot S-Model 5 is a 6-axis articulated robot specially developed for light parts handling with 5 kg payload. The mechanical unit construction is to have high rigidity and easy to maintain. This robot is controlled through an R-G controller unit with digital servo systems. The robot documentation can be found in the Robotics Lab. Requested: a. Study the FANUC robot documentation and specify the following: i. the robot components and their specifications ii. safety precautions iii. robot operation and control strategies iv. teaching methods and programming language. b. Using any software (Unigraphics, Matlab, Visual Basic, C++, Pascal,, etc), it is required to achieve the following tasks: i. make a complete solid modeling of this robot ii. animate all the robot joints in their working ranges iii. draw the robot working space in three views. c. Using the HTM Technique to obtain analytical expressions of robot models using the next steps: i. Draw the robot equivalent kinematic diagram ii. establish the orthonormal link coordinate systems (R i = o i, x i, y i, z i ) for i = to 7 indicating the base R b and tool coordinate system R t iii. define FANUC Denavit-Hartenberg (DH) parameters (a i, α i, r i, and θ i for i = to 6) iv. calculate the range of motion of each joint defined by (q i = θ i and θ i for i = to 6). v. define the elementary HTM's T i,i+ for such a robot for i = to 6. vi. calculate the robot HTM T 7 vii. define the robot DKPM viii. Make the transformation between the tool coordinates R t and the base coordinates R b ix. Try to operate the robot if possible. d. Each group must submit a Hard and Soft copies of a technical report (Max. pages) with its presentation (Max. slides) at most on Wednesday //5. Each member of the group must specify his role in the team on the report second page. This report must include the following items:. Abstract and objectives. Introduction 3. Results of (a, b and c) 4. Remarks and comments 5. References