Refer to Exercise 71 . (a) Suppose the wrist joint of the...

Refer to Exercise 71 . (a) Suppose the wrist joint of the robot's arm is allowed to rotate at the joint connection $S$ and the arm is located as shown in the first figure. The upper arm has a length of 15 inches; the forearm, without the hand, has a length of 10 inches; and the hand has a length of 7 inches. Approximate the coordinates of $R$ by using $\mathbf{a}+\mathbf{b}+\mathbf{c}$ (b). Suppose the robot's upper arm is rotated $75^{\circ},$ and then the forearm is rotated $-80^{\circ},$ and finally the hand is rotated an additional $40^{\circ},$ as shown in the second figure. Approximate the new coordinates of $R$ by using $\mathbf{d}+\mathbf{e}+\mathbf{f}$

Refer to Exercise 71 .
(a) Suppose the wrist joint of the robot's arm is allowed to rotate at the joint connection $S$ and the arm is located as shown in the first figure. The upper arm has a length of 15 inches; the forearm, without the hand, has a length of 10 inches; and the hand has a length of 7 inches. Approximate the coordinates of $R$ by using $\mathbf{a}+\mathbf{b}+\mathbf{c}$
(b). Suppose the robot's upper arm is rotated $75^{\circ},$ and then the forearm is rotated $-80^{\circ},$ and finally the hand is rotated an additional $40^{\circ},$ as shown in the second figure. Approximate the new coordinates of $R$ by using $\mathbf{d}+\mathbf{e}+\mathbf{f}$

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to form a resultant vector. In the context of robot arm positioning, each segment of the arm (upper arm, forearm, hand) can be represented as a vector, and the final position of the end effector is determined by adding these vectors together. This approach simplifies complex kinematic problems by breaking down the overall motion into the sum of individual segment motions.

Rotation Transformation

Rotation transformation involves changing the orientation of a vector in a coordinate system by applying a rotation. When an arm segment rotates, its representation as a vector is altered by using rotation formulas or matrices derived from trigonometric functions. This concept is crucial in determining how each segment's orientation affects its contribution to the overall position of the robot's arm.

Trigonometric Decomposition

Trigonometric decomposition is the method of breaking a vector into its horizontal and vertical components using sine and cosine functions. For each rotated segment of the robot arm, the component lengths in the x and y directions are calculated using trigonometry. This breakdown enables the computation of each segment’s contribution to the final coordinates when added together.

Forward Kinematics

Forward kinematics is a fundamental concept in robotics that deals with computing the end effector's position and orientation from given joint parameters such as angles and segment lengths. By sequentially applying vector addition, rotation transformations, and trigonometric decomposition, the overall position of the robot arm’s tip is determined, which is essential for tasks requiring precise movement.

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