Two joint arm to point control ------------------------------ .. figure:: https://github.com/AtsushiSakai/PythonRobotics/raw/master/ArmNavigation/two_joint_arm_to_point_control/animation.gif :alt: TwoJointArmToPointControl TwoJointArmToPointControl This is two joint arm to a point control simulation. This is a interactive simulation. You can set the goal position of the end effector with left-click on the ploting area. Inverse Kinematics for a Planar Two-Link Robotic Arm ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ A classic problem with robotic arms is getting the end-effector, the mechanism at the end of the arm responsible for manipulating the environment, to where you need it to be. Maybe the end-effector is a gripper and maybe you want to pick up an object and maybe you know where that object is relative to the robot - but you cannot tell the end-effector where to go directly. Instead, you have to determine the joint angles that get the end-effector to where you want it to be. This problem is known as inverse kinematics. Credit for this solution goes to: https://robotacademy.net.au/lesson/inverse-kinematics-for-a-2-joint-robot-arm-using-geometry/ First, let’s define a class to make plotting our arm easier. .. code-block:: ipython3 %matplotlib inline from math import cos, sin import numpy as np import matplotlib.pyplot as plt class TwoLinkArm: def __init__(self, joint_angles=[0, 0]): self.shoulder = np.array([0, 0]) self.link_lengths = [1, 1] self.update_joints(joint_angles) def update_joints(self, joint_angles): self.joint_angles = joint_angles self.forward_kinematics() def forward_kinematics(self): theta0 = self.joint_angles[0] theta1 = self.joint_angles[1] l0 = self.link_lengths[0] l1 = self.link_lengths[1] self.elbow = self.shoulder + np.array([l0*cos(theta0), l0*sin(theta0)]) self.wrist = self.elbow + np.array([l1*cos(theta0 + theta1), l1*sin(theta0 + theta1)]) def plot(self): plt.plot([self.shoulder[0], self.elbow[0]], [self.shoulder[1], self.elbow[1]], 'r-') plt.plot([self.elbow[0], self.wrist[0]], [self.elbow[1], self.wrist[1]], 'r-') plt.plot(self.shoulder[0], self.shoulder[1], 'ko') plt.plot(self.elbow[0], self.elbow[1], 'ko') plt.plot(self.wrist[0], self.wrist[1], 'ko') Let’s also define a function to make it easier to draw an angle on our diagram. .. code-block:: ipython3 from math import sqrt def transform_points(points, theta, origin): T = np.array([[cos(theta), -sin(theta), origin[0]], [sin(theta), cos(theta), origin[1]], [0, 0, 1]]) return np.matmul(T, np.array(points)) def draw_angle(angle, offset=0, origin=[0, 0], r=0.5, n_points=100): x_start = r*cos(angle) x_end = r dx = (x_end - x_start)/(n_points-1) coords = [[0 for _ in range(n_points)] for _ in range(3)] x = x_start for i in range(n_points-1): y = sqrt(r**2 - x**2) coords[0][i] = x coords[1][i] = y coords[2][i] = 1 x += dx coords[0][-1] = r coords[2][-1] = 1 coords = transform_points(coords, offset, origin) plt.plot(coords[0], coords[1], 'k-') Okay, we now have a TwoLinkArm class to help us draw the arm, which we’ll do several times during our derivation. Notice there is a method called forward_kinematics - forward kinematics specifies the end-effector position given the joint angles and link lengths. Forward kinematics is easier than inverse kinematics. .. code-block:: ipython3 arm = TwoLinkArm() theta0 = 0.5 theta1 = 1 arm.update_joints([theta0, theta1]) arm.plot() def label_diagram(): plt.plot([0, 0.5], [0, 0], 'k--') plt.plot([arm.elbow[0], arm.elbow[0]+0.5*cos(theta0)], [arm.elbow[1], arm.elbow[1]+0.5*sin(theta0)], 'k--') draw_angle(theta0, r=0.25) draw_angle(theta1, offset=theta0, origin=[arm.elbow[0], arm.elbow[1]], r=0.25) plt.annotate("$l_0$", xy=(0.5, 0.4), size=15, color="r") plt.annotate("$l_1$", xy=(0.8, 1), size=15, color="r") plt.annotate(r"$\theta_0$", xy=(0.35, 0.05), size=15) plt.annotate(r"$\theta_1$", xy=(1, 0.8), size=15) label_diagram() plt.annotate("Shoulder", xy=(arm.shoulder[0], arm.shoulder[1]), xytext=(0.15, 0.5), arrowprops=dict(facecolor='black', shrink=0.05)) plt.annotate("Elbow", xy=(arm.elbow[0], arm.elbow[1]), xytext=(1.25, 0.25), arrowprops=dict(facecolor='black', shrink=0.05)) plt.annotate("Wrist", xy=(arm.wrist[0], arm.wrist[1]), xytext=(1, 1.75), arrowprops=dict(facecolor='black', shrink=0.05)) plt.axis("equal") plt.show() .. image:: Planar_Two_Link_IK_files/Planar_Two_Link_IK_5_0.png It’s common to name arm joints anatomically, hence the names shoulder, elbow, and wrist. In this example, the wrist is not itself a joint, but we can consider it to be our end-effector. If we constrain the shoulder to the origin, we can write the forward kinematics for the elbow and the wrist. | :math:`elbow_x = l_0\cos(\theta_0)` | :math:`elbow_y = l_0\sin(\theta_0)` | :math:`wrist_x = elbow_x + l_1\cos(\theta_0+\theta_1) = l_0\cos(\theta_0) + l_1\cos(\theta_0+\theta_1)` | :math:`wrist_y = elbow_y + l_1\sin(\theta_0+\theta_1) = l_0\sin(\theta_0) + l_1\sin(\theta_0+\theta_1)` Since the wrist is our end-effector, let’s just call its coordinates \ :math:`x`\ and \ :math:`y`\ . The forward kinematics for our end-effector is then | :math:`x = l_0\cos(\theta_0) + l_1\cos(\theta_0+\theta_1)` | :math:`y = l_0\sin(\theta_0) + l_1\sin(\theta_0+\theta_1)` A first attempt to find the joint angles :math:`\theta_0` and :math:`\theta_1` that would get our end-effector to the desired coordinates :math:`x` and :math:`y` might be to try solving the forward kinematics for :math:`\theta_0` and :math:`\theta_1`, but that would be the wrong move. An easier path involves going back to the geometry of the arm. .. code-block:: ipython3 from math import pi arm.plot() label_diagram() plt.plot([0, arm.wrist[0]], [0, arm.wrist[1]], 'k--') plt.plot([arm.wrist[0], arm.wrist[0]], [0, arm.wrist[1]], 'b--') plt.plot([0, arm.wrist[0]], [0, 0], 'b--') plt.annotate("$x$", xy=(0.6, 0.05), size=15, color="b") plt.annotate("$y$", xy=(1, 0.2), size=15, color="b") plt.annotate("$r$", xy=(0.45, 0.9), size=15) plt.annotate(r"$\alpha$", xy=(0.75, 0.6), size=15) alpha = pi-theta1 draw_angle(alpha, offset=theta0+theta1, origin=[arm.elbow[0], arm.elbow[1]], r=0.1) plt.axis("equal") plt.show() .. image:: Planar_Two_Link_IK_files/Planar_Two_Link_IK_7_0.png The distance from the end-effector to the robot base (shoulder joint) is :math:`r` and can be written in terms of the end-effector position using the Pythagorean Theorem. :math:`r^2` = :math:`x^2 + y^2` Then, by the law of cosines, :math:`r`\ 2 can also be written as: :math:`r^2` = :math:`l_0^2 + l_1^2 - 2l_0l_1\cos(\alpha)` Because :math:`\alpha` can be written as :math:`\pi - \theta_1`, we can relate the desired end-effector position to one of our joint angles, :math:`\theta_1`. :math:`x^2 + y^2` = :math:`l_0^2 + l_1^2 - 2l_0l_1\cos(\alpha)` :math:`x^2 + y^2` = :math:`l_0^2 + l_1^2 - 2l_0l_1\cos(\pi-\theta_1)` :math:`2l_0l_1\cos(\pi-\theta_1) = l_0^2 + l_1^2 - x^2 - y^2` | :math:`\cos(\pi-\theta_1) = \frac{l_0^2 + l_1^2 - x^2 - y^2}{2l_0l_1}` | :math:`~` | :math:`~` | :math:`\cos(\pi-\theta_1) = -cos(\theta_1)` is a trigonometric identity, so we can also write :math:`\cos(\theta_1) = \frac{x^2 + y^2 - l_0^2 - l_1^2}{2l_0l_1}` which leads us to an equation for :math:`\theta_1` in terms of the link lengths and the desired end-effector position! :math:`\theta_1 = \cos^{-1}(\frac{x^2 + y^2 - l_0^2 - l_1^2}{2l_0l_1})` This is actually one of two possible solutions for :math:`\theta_1`, but we’ll ignore the other possibility for now. This solution will lead us to the “arm-down” configuration of the arm, which is what’s shown in the diagram. Now we’ll derive an equation for :math:`\theta_0` that depends on this value of :math:`\theta_1`. .. code-block:: ipython3 from math import atan2 arm.plot() plt.plot([0, arm.wrist[0]], [0, arm.wrist[1]], 'k--') p = 1 + cos(theta1) plt.plot([arm.elbow[0], p*cos(theta0)], [arm.elbow[1], p*sin(theta0)], 'b--', linewidth=5) plt.plot([arm.wrist[0], p*cos(theta0)], [arm.wrist[1], p*sin(theta0)], 'b--', linewidth=5) beta = atan2(arm.wrist[1], arm.wrist[0])-theta0 draw_angle(beta, offset=theta0, r=0.45) plt.annotate(r"$\beta$", xy=(0.35, 0.35), size=15) plt.annotate("$r$", xy=(0.45, 0.9), size=15) plt.annotate(r"$l_1sin(\theta_1)$",xy=(1.25, 1.1), size=15, color="b") plt.annotate(r"$l_1cos(\theta_1)$",xy=(1.1, 0.4), size=15, color="b") label_diagram() plt.axis("equal") plt.show() .. image:: Planar_Two_Link_IK_files/Planar_Two_Link_IK_9_0.png Consider the angle between the displacement vector :math:`r` and the first link :math:`l_0`; let’s call it :math:`\beta`. If we extend the first link to include the component of the second link in the same direction as the first, we form a right triangle with components :math:`l_0+l_1cos(\theta_1)` and :math:`l_1sin(\theta_1)`, allowing us to express :math:`\beta` as :math:`\beta = \tan^{-1}(\frac{l_1\sin(\theta_1)}{l_0+l_1\cos(\theta_1)})` We now have an expression for this angle :math:`\beta` in terms of one of our arm’s joint angles. Now, can we relate :math:`\beta` to :math:`\theta_0`? Yes! .. code-block:: ipython3 arm.plot() label_diagram() plt.plot([0, arm.wrist[0]], [0, arm.wrist[1]], 'k--') plt.plot([arm.wrist[0], arm.wrist[0]], [0, arm.wrist[1]], 'b--') plt.plot([0, arm.wrist[0]], [0, 0], 'b--') gamma = atan2(arm.wrist[1], arm.wrist[0]) draw_angle(beta, offset=theta0, r=0.2) draw_angle(gamma, r=0.6) plt.annotate("$x$", xy=(0.7, 0.05), size=15, color="b") plt.annotate("$y$", xy=(1, 0.2), size=15, color="b") plt.annotate(r"$\beta$", xy=(0.2, 0.2), size=15) plt.annotate(r"$\gamma$", xy=(0.6, 0.2), size=15) plt.axis("equal") plt.show() .. image:: Planar_Two_Link_IK_files/Planar_Two_Link_IK_12_0.png Our first joint angle :math:`\theta_0` added to :math:`\beta` gives us the angle between the positive :math:`x`-axis and the displacement vector :math:`r`; let’s call this angle :math:`\gamma`. :math:`\gamma = \theta_0+\beta` :math:`\theta_0`, our remaining joint angle, can then be expressed as :math:`\theta_0 = \gamma-\beta` We already know :math:`\beta`. :math:`\gamma` is simply the inverse tangent of :math:`\frac{y}{x}`, so we have an equation of :math:`\theta_0`! :math:`\theta_0 = \tan^{-1}(\frac{y}{x})-\tan^{-1}(\frac{l_1\sin(\theta_1)}{l_0+l_1\cos(\theta_1)})` We now have the inverse kinematics for a planar two-link robotic arm. If you’re planning on implementing this in a programming language, it’s best to use the atan2 function, which is included in most math libraries and correctly accounts for the signs of :math:`y` and :math:`x`. Notice that :math:`\theta_1` must be calculated before :math:`\theta_0`. | :math:`\theta_1 = \cos^{-1}(\frac{x^2 + y^2 - l_0^2 - l_1^2}{2l_0l_1})` | :math:`\theta_0 = atan2(y, x)-atan2(l_1\sin(\theta_1), l_0+l_1\cos(\theta_1))`