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17.1.1 Degrees of Freedom

• The scalar and vector approaches are easily extended to 3D problems. One significant difference is that the polar notations are no longer available for use.

• We can determine the number of degrees of freedom using a simple relationship that is an extension of the Kutzbach criteria,

• Consider the number of degrees of freedom in the linkage below,

### 17.2 Homogeneous Matrices

• This method still uses geometry to determine the position of the robot, but it is put into an ordered method using matrices.

• Consider the planar robot below,

• The basic approach to this method is,

1. On the base, each joint, and the tool of the robot, attach a reference frame (most often x-y-z). Note that the last point is labels ‘T’ for tool. This will be a convention that I will generally follow.

2. Determine a transformation matrix to map between each frame. It is important to do this by assuming the joints are in their 0 joint positions. Put the joint positions in as variables.

3. Multiply the frames to get a complete transformation matrix.

• The position and orientation can be read directly from the homogenous transformation matrix as indicated above.

• To reverse the transform, we only need to invert the transform matrix - this is a direct result of the loop equation.

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17.4.2 Newton-Euler

• We can sum forces and moments, and then solve the equations in a given sequence.

• These equations can be written in vector form,

• To do these calculations start at the base, and calculate the kinematics up to the end of the manipulator (joint positions, velocities and accelerations). Then work back from the end and find forces and moments.

### 17.5 References

Erdman, A.G. and Sandor, G.N., Mechanism Design Analysis and Synthesis, Vol. 1, 3rd Edition, Prentice Hall, 1997.

Fu, Gonzalez, and Lee,

Shigley, J.E., Uicker, J.J., “Theory of Machines and Mechanisms, Second Edition, McGraw-Hill, 1995.

### 17.6 Practice Problems

1. For the Stanford arm below,

a) list the D-H parameters (Hint: extra “dummy” joints may be required)

b) Find the forward kinematics using homogenous matrices.

c) Find the Jacobian matrix for the arm.

d) If the arm is at θ1 = 45 degrees, θ2 = 45 degrees, r = 0.5m, find the speed of the TCP if the joint velocities are θ’1 = 1 degree/sec, θ’2 = 10 degrees/sec, and r’ = 0.01 m/sec.

3. Robotics and Automated Manipulators (RAM) has consulted you about a new robotic manipulator. This work will include kinematic analysis, gears, and the tool. The robot is pictured below. The robot is shown on the next page in the undeformed position. The tool is a gripper (finger) type mechanism.

The robot is drawn below in the undeformed position. The three positioning joints are shown, and a frame at the base and tool are also shown.

The tool is a basic gripper mechanism, and is shown as a planar mechanism below. As the cylinder moves to the left the fingers close.

a) The first thing you do is determine what sequence of rotations and translations are needed to find the tool position relative to the base position.

b) As normal, you decide to relate a cartesian (x-y) velocity of the gripper to joint velocities. Set up the calculation steps needed to do this based on the results in question #1.

c) To drive the revolute joints RAM has already selected two similar motors that have a maximum velocity. You decide to use the equations in question #2, with maximum specified tool velocities to find maximum joint velocities. Assume that helical gears are to be used to drive the revolute joints, specify the basic dimensions (such as base circle dia.). List the steps to develop the geometry of the gears, including equations.

d) The gripper fingers may close quickly, and as a result a dynamic analysis is deemed necessary. List the steps required to do an analysis (including equations) to find the dynamic forces on the fingers.

e) The idea of using a cam as an alternate mechanism is being considered. Develop a design that is equivalent to the previous design. Sketch the mechanism and a detailed displacement graph of the cam.

f) The sliding joint ‘r’ has not been designed yet. RAM wants to drive the linear motion, without using a cylinder. Suggest a reasonable design, and sketch.

4. For an articulated robot, find the forward, and inverse kinematics using geometry, homogenous matrices, and Denavit-Hartenberg transformations.

5. Assign Denavit-Hartenberg link parameters to an articulated robot.

6. For the Stanford arm below,

a) list the D-H parameters (Hint: extra “dummy” joints may be required)

b) Find the forward kinematics using homogenous matrices.

c) Find the Jacobian matrix for the arm.

d) If the arm is at θ1 = 45 degrees, θ2 = 45 degrees, r = 0.5m, find the speed of the TCP if the joint velocities are θ’1 = 1 degree/sec, θ’2 = 10 degrees/sec, and r’ = 0.01 m/sec.

7. Consider the forward kinematic transformation of the two link manipulator below. Given the position of the joints, and the lengths of the links, determine the location of the tool centre point using a) basic geometry, b) homogenous transforms, and c) Denavit-Hartenberg transformations.

a) For the robot described in question 1 determine the inverse kinematics for the robot. (i.e., given the position of the tool, determine the joint angles of the robot.) Keep in mind that in this case the solution will have two different cases. Determine two different sets of joint angles required to position the TCP at x=5”, y=6”.

b) For the inverse kinematics of question #2, what conditions would indicate the robot position is unreachable? Are there any other cases that are indeterminate?

8 Find the dynamic forces in the system below,

9. Examine the robot figure below and,

a) assign frames to the appropriate joints.

b) list the transformations for the forward kinematics.

c) expand the transformations to matrices (do not multiply).

10. Given the transformation matrix below for a polar robot,

a) find the Jacobian matrix.

b) Given the joint positions, find the forward and inverse Jacobian matrices.

c) If we are at the position below, and want to move the tool at the given speed, what joint velocities are required?

11. Examine the robot figure below and,

a) assign frames to the appropriate joints.

b) list the transformations for the forward kinematics.

c) expand the transformations to matrices (do not multiply).

12. Given the transformation matrix below for a polar robot,

a) find the Jacobian matrix.

b) Given the joint positions, find the forward and inverse Jacobian matrices.

c) If we are at the position below, and want to move the tool at the given speed, what joint velocities are required?

13. Find the forward kinematics for the robots below using homogeneous and Denavit-Hartenberg matrices.

14. Use the equations below to find the inverse Jacobian. Use the inverse Jacobian to find the joint velocities required at t=0.5s.