# Step-by-Step Solution: Kinematics of a Three-Link RRP Manipulator ## 1. **Understanding the RRP Manipulator Structure** The manipulator in the image is a **Three-link RRP (Revolute-Revolute-Prismatic)** manipulator: - **R**: Revolute joint (rotation, θ₂) - **R**: Revolute joint (rotation, θ₃) - **P**: Prismatic joint (linear, d₁) - **L₁, L₂**: Link lengths ## 2. **Kinematic Equations (Forward Kinematics)** We need to determine the end-effector position \((x, y, z)\) in terms of the joint variables and link lengths. ### **Assigning Frames (DH Convention)** Let's assign the frames as per the Denavit-Hartenberg (DH) convention: - **Joint 1 (Prismatic, d₁):** moves the arm **up/down** along the Z-axis. - **Joint 2 (Revolute, θ₂):** rotates around the Z-axis. - **Joint 3 (Revolute, θ₃):** rotates around its own axis (after θ₂). ### **Position of End-Effector** The position of the end-effector relative to the base is: \[ \begin{align*} x &= (L_1 + L_2) \cos \theta_2 \cos \theta_3 \\ y &= (L_1 + L_2) \sin \theta_2 \cos \theta_3 \\ z &= d_1 + (L_1 + L_2) \sin \theta_3 \end{align*} \] But, for this typical RRP structure (based on the image), the kinematic equations are: \[ \begin{align*} x &= (L_1 \cos \theta_2 + L_2 \cos (\theta_2 + \theta_3)) \\ y &= (L_1 \sin \theta_2 + L_2 \sin (\theta_2 + \theta_3)) \\ z &= d_1 \end{align*} \] --- ## 3. **Plug in Given Values** Given: - \(\theta_2 = 30^\circ\) - \(\theta_3 = 45^\circ\) - \(d_1 = 15\) - \(L_1 = 25\) - \(L_2 = 25\) Convert angles to radians for calculation: - \(\theta_2 = 30^\circ = \frac{\pi}{6}\) radians - \(\theta_3 = 45^\circ = \frac{\pi}{4}\) radians Calculate \(\theta_2 + \theta_3\): \[ \theta_2 + \theta_3 = 30^\circ + 45^\circ = 75^\circ = 1.308996938 \text{ radians} \] --- ## 4. **Calculate Each Term** ### **Calculate cos and sin values** - \(\cos 30^\circ = 0.8660\) - \(\sin 30^\circ = 0.5\) - \(\cos 75^\circ = 0.2588\) - \(\sin 75^\circ = 0.9659\) --- ### **Plug into Equations** \[ \begin{align*} x &= 25 \cdot 0.8660 + 25 \cdot 0.2588 \\ y &= 25 \cdot 0.5 + 25 \cdot 0.9659 \\ z &= 15 \\ \end{align*} \] --- ### **Compute Each Value** #### **For x:** \[ x = 25 \times 0.8660 + 25 \times 0.2588 = 21.65 + 6.47 = 28.12 \] #### **For y:** \[ y = 25 \times 0.5 + 25 \times 0.9659 = 12.5 + 24.15 = 36.65 \] #### **For z:** \[ z = 15 \] --- ## 5. **Final Answer** \[ \boxed{ (x, y, z) = (28.12,\ 36.65,\ 15) } \] --- ## 6. **Summary Table** | Variable | Value | |-----------------------|-------------------------| | \(\theta_2\) | \(30^\circ\) | | \(\theta_3\) | \(45^\circ\) | | \(d_1\) | \(15\) | | \(L_1\), \(L_2\) | \(25\), \(25\) | | \(x\) | \(28.12\) | | \(y\) | \(36.65\) | | \(z\) | \(15\) | --- ## 7. **Explanation** - The **\(x\) and \(y\)** positions are found by projecting the link lengths along the directions determined by the two angles. - The **\(z\)** position is simply the prismatic joint extension. If you have more questions or want to see the transformation matrices, let me know!