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Problem 1. (16 points) We are given an n-dof manipulator, where each link i, i = 1,..., n, is assigned a body-fixed reference frame {i} defined using the Denavit-Hartenberg convention. The fixed base frame is denoted by {0}, whereas the end-effector frame, denoted by {n+1}, is always aligned with frame {n}. 1. Consider the case of two adjacent revolute joints, labeled i - 1 and i, respectively, as shown in Figure 1. ai-1 axis i 1 axis i Zi-1 i-1 ai-1 \Xi-1 - It Figure 1: Denavit-Hartenberg convention for two adjacent revolute joints. The given Denavit-Hartenberg parameters are -1 = const, a-1 = const, di = const, whereas oi = Oi, where ; is the joint variable for joint {i}. Find the expression of the screw axis of joint i resolved in the frame {i-1}, denoted by Si-i,i R6, when Oi = = 0. 2. Consider the case of a revolute joint adjacent to a prismatic joint, labeled i - 1 and i, respectively, as shown in Figure 2. axis i 1 Zi-1 i-1 ai-1 i-1 Xi-1 di axis i |||- Figure 2: Denavit-Hartenberg convention for a revolute joint adjacent to a prismatic joint. In this case, the given Denavit-Hartenberg parameters are ai-1 = const, ai-1 = const, = = const, whereas di Oi, where 0 is the joint variable for joint {i}. Find the expression of the screw axis of joint i resolved in the frame {i-1}, denoted by Si-ii ER6, when 0 = 0. 3. On the basis of the results obtained so far, compute the forward kinematics T(0) for the 4-dof manipulator whose DH-parameters are listed in Table 1, using the spatial formulation of the product of exponentials formula T(0) = [S1]01 [S2]02S303S404 M where Si = So,i, i = 1,...,n, is the expression of the screw axis of the i-th joint resolved in the base frame {0} at the home position, and M := T(0). Note: You only need to compute Si, i = 1,..., 4, and M, not the matrix exponentials. 2 ai-1 ai-1 di oi 1 -90 0 0 01 2 180 Li 0 02 3 90 0 03 0 4 0 L2 L3 04 Table 1: Denavit-Hartenberg parameters for a 4-dof manipulator 4. Compute the forward kinematics T'(0) for the 4-dof manipulator whose DH-parameters are listed in Table 1, using the body formulation of the product of exponentials formula T'(0) = Me[B]01B202B303B404 where Bi = Sn+1,i, i = 1,...,n, is the expression of the screw axis of the i-th joint resolved in the end-effector frame {n+1} at the home position, and M := = T'(0). Note: You only need to compute Bi, i = 1,..., 4, and M, not the matrix exponentials. Problem 1. (16 points) We are given an n-dof manipulator, where each link i, i = 1,..., n, is assigned a body-fixed reference frame {i} defined using the Denavit-Hartenberg convention. The fixed base frame is denoted by {0}, whereas the end-effector frame, denoted by {n+1}, is always aligned with frame {n}. 1. Consider the case of two adjacent revolute joints, labeled i - 1 and i, respectively, as shown in Figure 1. ai-1 axis i 1 axis i Zi-1 i-1 ai-1 \Xi-1 - It Figure 1: Denavit-Hartenberg convention for two adjacent revolute joints. The given Denavit-Hartenberg parameters are -1 = const, a-1 = const, di = const, whereas oi = Oi, where ; is the joint variable for joint {i}. Find the expression of the screw axis of joint i resolved in the frame {i-1}, denoted by Si-i,i R6, when Oi = = 0. 2. Consider the case of a revolute joint adjacent to a prismatic joint, labeled i - 1 and i, respectively, as shown in Figure 2. axis i 1 Zi-1 i-1 ai-1 i-1 Xi-1 di axis i |||- Figure 2: Denavit-Hartenberg convention for a revolute joint adjacent to a prismatic joint. In this case, the given Denavit-Hartenberg parameters are ai-1 = const, ai-1 = const, = = const, whereas di Oi, where 0 is the joint variable for joint {i}. Find the expression of the screw axis of joint i resolved in the frame {i-1}, denoted by Si-ii ER6, when 0 = 0. 3. On the basis of the results obtained so far, compute the forward kinematics T(0) for the 4-dof manipulator whose DH-parameters are listed in Table 1, using the spatial formulation of the product of exponentials formula T(0) = [S1]01 [S2]02S303S404 M where Si = So,i, i = 1,...,n, is the expression of the screw axis of the i-th joint resolved in the base frame {0} at the home position, and M := T(0). Note: You only need to compute Si, i = 1,..., 4, and M, not the matrix exponentials. 2 ai-1 ai-1 di oi 1 -90 0 0 01 2 180 Li 0 02 3 90 0 03 0 4 0 L2 L3 04 Table 1: Denavit-Hartenberg parameters for a 4-dof manipulator 4. Compute the forward kinematics T'(0) for the 4-dof manipulator whose DH-parameters are listed in Table 1, using the body formulation of the product of exponentials formula T'(0) = Me[B]01B202B303B404 where Bi = Sn+1,i, i = 1,...,n, is the expression of the screw axis of the i-th joint resolved in the end-effector frame {n+1} at the home position, and M := = T'(0). Note: You only need to compute Bi, i = 1,..., 4, and M, not the matrix exponentials.