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- Question : 1E - Using the methods of Section 2.1 derive a formula, in terms of n, for the number of degrees of freedom of a rigid body in n-dimensional space. Indicate how many of these dof are translational and how many are rotational. Describe the topology of the C-space
- Question : 2E - In the previous exercise, we assumed that your arm is a serial chain. In fact, between your upper arm bone (the humerus) and the bone complex just past your wrist (the carpal bones), your forearm has two bones, the radius and the ulna, which are part of a closed chain. Model your arm, from your shoulder to your palm, as a mechanism with joints and calculate the number of degrees of freedom using Gr
- Question : 3E - Assume each of your arms has n degrees of freedom. You are driving a car, your torso is stationary relative to the car (owing to a tight seatbelt!), and both hands are firmly grasping the wheel, so that your hands do not move relative to the wheel. How many degrees of freedom does your arms-plus-steering wheel system have? Explain your answer.
- Question : 4E - Figure 2.15 shows a robot used for human arm rehabilitation. Determine the number of degrees of freedom of the chain formed by the human arm and the robot
- Question : 5E - Consider a spatial parallel mechanism consisting of a moving plate connected to a fixed plate by n identical legs. For the moving plate to have six degrees of freedom, how many degrees of freedom should each leg have, as a function of n? For example, if n = 3 then the moving plate and fixed plate are connected by three legs; how many degrees of freedom should each leg have for the moving plate to move with six degrees of freedom? Solve for arbitrary n.
- Question : 6E - Use the planar version of Gr
- Question : 7E - Use the planar version of Gr
- Question : 8E - Use the spatial version of Gr
- Question : 9E - Figure 2.25 shows a spherical four-bar linkage, in which four links (one of the links is the ground link) are connected by four revolute joints to form a single-loop closed chain. The four revolute joints are arranged so that they lie on a sphere such that their joint axes intersect at a common point. (a) Use Gr
- Question : 10E - Figure 2.27 shows a 3?PUP platform, in which three identical PUP legs connect a fixed base to a moving platform. The P joints on both the fixed base and moving platform are arranged symmetrically. Use Gr
- Question : 11E - The dual-arm robot of Figure 2.28 is rigidly grasping a box. The box can only slide on the table; the bottom face of the box must always be in contact with the table. How many degrees of freedom does this system have?
- Question : 12E - A caterpillar robot. (a) A caterpillar robot is hanging by its tail end as shown in Figure 2.30(a). The robot consists of eight serially connected rigid links (one head, one tail, and six body links). The six body links are connected by revolute
- Question : 13E - The workspace of a planar 3R open chain. (a) Consider a planar 3R open chain with link lengths (starting from the fixed base joint) 5, 2, and 1, respectively. Considering only the Cartesian point of the tip, draw its workspace. (b) Now consider a planar 3R open chain with link lengths (starting from the fixed base joint) 1, 2, and 5, respectively. Considering only the Cartesian point of the tip, draw its workspace. Which of these two chains has a larger workspace? (c) A not-so-clever designer claims that he can make the workspace of any planar open chain larger simply by increasing the length of the last link. Explain the fallacy behind this claim.
- Question : 14E - Task space. (a) Describe the task space for a robot arm writing on a blackboard. (b) Describe the task space for a robot arm twirling a baton.
- Question : 15E - Describe an algorithm that drives the rolling coin of Figure 2.11 from any arbitrary initial configuration in its four-dimensional C-space to any arbitrary goal configuration, despite the two nonholonomic constraints. The control inputs are the rolling speed ? ? and the turning speed ?$. You should explain clearly in words or pseudocode how the algorithm would work. It is not necessary to give actual code or formulas.
- Question : 16E - A di?erential-drive mobile robot has two wheels that do not steer but whose speeds can be controlled independently. The robot goes forward and backward by spinning the wheels in the same direction at the same speed, and it turns by spinning the wheels at di?erent speeds. The configuration of the robot is given by five variables: the (x, y) location of the point halfway between the wheels, the heading direction ? of the robot

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