Robotic Arm Path Planning (Mechanical & Aerospace Engineering) Solution

In robotics, the forward kinematics function maps the joint angles of a robotic arm to a physical coordinate $(x, y)$ in space. However, to control the robot, engineers must solve the inverse kinematics problem: given a target coordinate $(x, y)$, calculate the joint angle $\theta$ required to reach it.

For a single-link arm of length $L = 5\text{ units}$, the position is given by $x = 5\cos\theta$ and $y = 5\sin\theta$.

1. Reconstruct the trigonometric relationship to show that $\theta = \arctan\left(\frac{y}{x}\right)$.
2. If the arm needs to target the coordinate $(x, y) = (-3, 4)$ in Quadrant II, evaluate the joint angle $\theta$ in radians, accounting for quadrant corrections.

Step-by-Step Solution:

1. Using the given coordinates:
   $$\frac{y}{x} = \frac{5\sin\theta}{5\cos\theta} = \tan\theta$$
   Taking the inverse tangent of both sides:
   $$\theta = \arctan\left(\frac{y}{x}\right)$$
2. For $(x, y) = (-3, 4)$:
   $$\theta_{\text{ref}} = \arctan\left(\left|\frac{4}{-3}\right|\right) = \arctan\left(\frac{4}{3}\right) \approx 0.927\text{ rad}$$
   Since $x < 0$ and $y > 0$, the target lies in Quadrant II. The true angle $\theta$ is:
   $$\theta = \pi - \theta_{\text{ref}} = \pi - \arctan\left(\frac{4}{3}\right) \approx 3.142 - 0.927 = 2.215\text{ rad}$$

Meaning of the Answer & Real-Life Application:

Standard inverse trigonometric functions like $\arctan(z)$ are mathematically restricted to Quadrants I and IV ($(-\frac{\pi}{2}, \frac{\pi}{2})$). In robotics and path planning, software engineers use quadrant dynamics to reconstruct the true angle in 2D space (often using functions like atan2(y, x)) to prevent the arm from moving to the incorrect mirror-image location.