Let's solve the problem step by step: ### **Step 1: Understanding the Robot Configuration** - The robot is a planar manipulator with: - A **prismatic joint** (\(d_1\)) moving vertically. - Two **revolute joints** with 1 m links each. - The base is 2m above the ground. - The end-effector's position is given as \([0,\ 1.3,\ 4.5]\) in the base frame (Frame #0). ### **Step 2: Forward Kinematics** Let the coordinates of the end effector (EEF) be \((x, y, z)\). From the diagram: - The prismatic joint (\(d_1\)) moves along the **z-axis**. - The first link (1m) rotates, then a second link (1m), then a third link (1m, with angle \(\theta_2\)). - The **y-coordinate** is along the horizontal links. #### **Kinematic Chain** - The base is at \((0, 0, 2)\). - After prismatic joint: \((0, 0, 2 + d_1)\). - After 1st link (1m, straight up): \((0, 0, 2 + d_1 + 1)\). - After 2nd link (1m, straight up): \((0, 0, 2 + d_1 + 2)\). - After 3rd link (1m, angle \(\theta_2\)): This link can move in the \(y_0\)-\(z_0\) plane. So, **end effector position:** \[ y = 1 + 1 \cdot \sin(\theta_2) \\ z = 2 + d_1 + 2 + 1 \cdot \cos(\theta_2) \] But from the diagram, the last two links (after prismatic) are both 1m, then the last link is at an angle \(\theta_2\). So, the vertical position is: \[ z = 2 + d_1 + 1 + 1 \cdot \cos(\theta_2) \] And horizontal position: \[ y = 1 + 1 \cdot \sin(\theta_2) \] But looking at the image, the two links are straight and the third link (1m) is at angle \(\theta_2\). So, after extending prismatic joint \(d_1\), we go up 1m, then up another 1m (total 2m after \(d_1\)), then the last link at angle \(\theta_2\). So, **the position of the end-effector:** \[ y = 1 \cdot \sin(\theta_2) \] \[ z = 2 + d_1 + 2 + 1 \cdot \cos(\theta_2) \] But from the diagram, after \(d_1\), the robot extends **up 2m** (from the two links), then the last link at \(\theta_2\). So, - \(z = 2 + d_1 + 2 + 1\cdot\cos(\theta_2)\) - \(y = 1\cdot\sin(\theta_2)\) But **the base starts at 2m**, then adds \(d_1\), then two 1m links (so \(2 + d_1 + 2\)), then the last link at angle \(\theta_2\). Thus, \[ z = 2 + d_1 + 2 + 1\cdot\cos(\theta_2) = 4 + d_1 + \cos(\theta_2) \] \[ y = 1\cdot\sin(\theta_2) \] ### **Step 3: Plug in End-Effector Position** Given: \([0, 1.3, 4.5]\) So, \[ y = 1.3 = 1\cdot \sin(\theta_2) \implies \sin(\theta_2) = 1.3 \] But this is not possible (\(|\sin(\theta_2)| \leq 1\)). Let's check the image again. #### Correction: Looking at the diagram, the **sum of the two links plus the last link** gives the y position. Let the last joint angle be \(\theta_2\) (with respect to the previous vertical position). So: - \(y = 1 + 1 + 1\cdot\sin(\theta_2) = 2 + \sin(\theta_2)\) - \(z = 2 + d_1 + 1\cdot\cos(\theta_2)\) Given: \(y = 1.3\), \(z = 4.5\) So, #### **Equation System** \[ y: \quad 1.3 = 2 + \sin(\theta_2) \implies \sin(\theta_2) = 1.3 - 2 = -0.7 \] \[ z: \quad 4.5 = 2 + d_1 + \cos(\theta_2) \implies d_1 = 4.5 - 2 - \cos(\theta_2) = 2.5 - \cos(\theta_2) \] ### **Step 4: Solve for \(\theta_2\)** \(\sin(\theta_2) = -0.7\) \(\theta_2 = \arcsin(-0.7)\) - \(\theta_2 \approx -44.4^\circ\) or \(180^\circ + (-44.4^\circ) = 135.6^\circ\) Now, \(\cos(\theta_2)\) for both solutions: - For \(\theta_2 = -44.4^\circ\): - \(\cos(-44.4^\circ) = \cos(44.4^\circ) \approx 0.715\) - \(d_1 = 2.5 - 0.715 = 1.785\) - For \(\theta_2 = 135.6^\circ\): - \(\cos(135.6^\circ) \approx -0.715\) - \(d_1 = 2.5 - (-0.715) = 3.215\) But these values **do not match any of the options**. Let's check the calculations. #### **Check if the y-coordinate is correct** From the diagram, after the prismatic joint, the two 1m links are **vertical**; only the last link has a y-component. So, \[ y = 1 \cdot \sin(\theta_2) \] So, \[ 1.3 = \sin(\theta_2) \implies \text{Not possible.} \] Let's try with the sum of all three links: - \(y = 1 + 1 + 1\cdot\sin(\theta_2)\), as above. But then, \(\sin(\theta_2) = 1.3 - 2 = -0.7\). So, let's check which option matches \(\sin(\theta_2) \approx -0.7\): - \(\theta_2 = -44.4^\circ\) or \(224.4^\circ\). Let's check the options: #### Option Calculations 1. \(d_1 = 0.671, \theta_2 = -34^\circ\) - \(\sin(-34^\circ) \approx -0.559\) (no) 2. \(d_1 = 0.568, \theta_2 = 56^\circ\) - \(\sin(56^\circ) \approx 0.829\) (no) 3. \(d_1 = 2.454, \theta_2 = 72.5^\circ\) - \(\sin(72.5^\circ) \approx 0.951\) (no) 4. \(d_1 = 0.314, \theta_2 = -68^\circ\) - \(\sin(-68^\circ) \approx -0.927\) (no) 5. \(d_1 = 0.546, \theta_2 = -72.5^\circ\) - \(\sin(-72.5^\circ) \approx -0.951\) (no) 6. \(d_1 = 2.74, \theta_2 = 68^\circ\) - \(\sin(68^\circ) \approx 0.927\) (no) None match exactly, but the closest to \(-0.7\) is option 1 (\(-0.559\)) and option 4 (\(-0.927\)). Let's check the corresponding \(d_1\) values for each: For Option 1: \(\theta_2 = -34^\circ\) - \(\cos(-34^\circ) \approx 0.829\) - \(d_1 = 2.5 - 0.829 = 1.671\) (but option gives 0.671) - So, not matching. For Option 4: \(\theta_2 = -68^\circ\) - \(\cos(-68^\circ) \approx 0.374\) - \(d_1 = 2.5 - 0.374 = 2.126\) (but option gives 0.314) - Not matching. Let's check the calculation again using the **z-coordinate** for all options. ### **Step 5: Test All Options** Given \(z = 4.5\), \(z = 2 + d_1 + \cos(\theta_2)\): #### Option 1: - \(d_1 = 0.671\), \(\theta_2 = -34^\circ\), \(\cos(-34^\circ) = 0.829\) - \(2 + 0.671 + 0.829 = 3.5\) (not matching 4.5) #### Option 2: - \(d_1 = 0.568\), \(\theta_2 = 56^\circ\), \(\cos(56^\circ) = 0.559\) - \(2 + 0.568 + 0.559 = 3.127\) (not matching 4.5) #### Option 3: - \(d_1 = 2.454\), \(\theta_2 = 72.5^\circ\), \(\cos(72.5^\circ) = 0.303\) - \(2 + 2.454 + 0.303 = 4.757\) (close to 4.5) #### Option 4: - \(d_1 = 0.314\), \(\theta_2 = -68^\circ\), \(\cos(-68^\circ) = 0.374\) - \(2 + 0.314 + 0.374 = 2.688\) (not matching 4.5) #### Option 5: - \(d_1 = 0.546\), \(\theta_2 = -72.5^\circ\), \(\cos(-72.5^\circ) = 0.303\) - \(2 + 0.546 + 0.303 = 2.849\) (not matching 4.5) #### Option 6: - \(d_1 = 2.74\), \(\theta_2 = 68^\circ\), \(\cos(68^\circ) = 0.374\) - \(2 + 2.74 + 0.374 = 5.114\) (not matching 4.5) ### **Step 6: Test Y-coordinate for Closest z-match** Option 3: \(d_1 = 2.454\), \(\theta_2 = 72.5^\circ\) - \(\sin(72.5^\circ) = 0.951\) - \(y = 0.951\) (not matching 1.3) Option 6: \(d_1 = 2.74\), \(\theta_2 = 68^\circ\) - \(\sin(68^\circ) = 0.927\) - \(y = 0.927\) (not matching 1.3) But none of the options exactly match both values. ### **Step 7: Final Check** With the given options, the one **closest to the required z-coordinate (4.5)** is **Option 3** (\(z = 4.757\), \(y = 0.951\)), and **Option 6** (\(z = 5.114\), \(y = 0.927\)). These options are the closest possible with the given choices. ### **Final Answer** **Check the boxes:** - \(\boxed{d_1 = 2.454,\ \theta_2 = 72.5^\circ}\) (Closest match to the given end-effector coordinates with the available answer choices.) --- **Summary of Reasoning:** - Used forward kinematics based on the robot's structure. - Set up equations for y and z based on link lengths and joint variables. - Solved and compared to answer choices. - Selected the closest valid answer. --- **If you have to choose one or more, select:** - \(d_1 = 2.454,\ \theta_2 = 72.5^\circ\)