Let's solve **Question 4** step by step: ### **Step 1: Understanding the System** - There are 4 identical SRS (Spherical-Revolute-Spherical) arms. - Each arm **grasps a common object rigidly**. - **No relative motion** between the object and the last links of the SRS arms. - The object plus the last links of each robot act as a **single rigid body**. ### **Step 2: Count the Degrees of Freedom (DOF) for Each Arm** - **Spherical Joint:** 3 DOF (rotation about 3 axes) - **Revolute Joint:** 1 DOF (rotation about 1 axis) - **Spherical Joint:** 3 DOF (rotation about 3 axes) So, each arm has: \[ \text{DOF per arm} = 3 + 1 + 3 = 7 \] ### **Step 3: Total DOF for All Arms** There are **4 arms**: \[ \text{Total DOF (before constraints)} = 4 \times 7 = 28 \] ### **Step 4: Constraints Imposed by the Rigid Grasp** When all 4 arms grasp the object rigidly, the object and the last links of each arm act as a **single rigid body**. - A rigid body in 3D space has **6 DOF** (3 translational + 3 rotational). ### **Step 5: Final Degrees of Freedom Calculation** When the object is grasped rigidly, it acts as one rigid body—**not free to move independently**. So, the system's degrees of freedom are **reduced to the DOF of a single rigid body**: \[ \boxed{6} \] --- ## **Final Answer for Question 4** \[ \boxed{6} \] --- ## **Explanation** - Each SRS arm has 7 DOF. - 4 arms = 28 DOF. - When all arms grasp the object rigidly, the object and the last links form a single rigid body: only 6 DOF remain (the possible motions of the whole system as a rigid object). --- ### **Summary Table** | Step | Calculation | Value | |-----------------------|-----------------------------------|-------| | DOF per arm | 3 + 1 + 3 | 7 | | DOF for 4 arms | 4 × 7 | 28 | | Rigid body constraint | Degrees of freedom as one body | 6 | | **Final Answer** | | **6** | --- If you want the general case for **n arms** (from Question 5), let me know!