### **I. Introduction** ![image.png](https://images.hive.blog/DQmSTDUW4WRXz4U8Dyfr9EneNjB48w6iZFasoTaEc1nnmda/image.png) The mathematical modeling of the kinematics of a 3R planar manipulator involved in identifying the end-effector position or the joint angles. The 3R planar manipulator has three revolute joint and three links, as shown in Figure 1. The robot forward kinematics yields the end-effector position and its orientation from the given link lengths and joint angles. On the contrary, the robot inverse kinematics finds the joint angles from the given end-effector and its orientation. The position of the robot end-effector is denoted by P as ![image.png](https://images.hive.blog/DQmZVMGpFyvMr3h2XNJZEPc37ES8o6BxyuM9aT9VxdEmzVz/image.png) and its orientation relative to the world x-axis is denoted *γ*. The robot joint angles are denoted θ_i. The trigonometric functions, sine and cosine, are used extensively in the text so the following notations are introduced. ![image.png](https://images.hive.blog/DQmX5MSdeamw58WD69p2wYuQ5NPuGcH4XnHKjQPqMea7dEN/image.png) This notation is also extended to sums as ![image.png](https://images.hive.blog/DQmNv8jJCyHXz4RkzKRtBnVLCu9nQg8FT2yvAuUmxhtLbsU/image.png) Link lengths are the distances between the joints and denoted as *L*_ij , where *i* is the joint number closer to the base and *j* is the joint number to the end-effector. In this text, we derived equations for the forward and inverse kinematics of a 3R planar manipulator, in accordance to the earlier notations. Also, we create *fkinematics* and *ikinematics* functions in MATLAB for forward and inverse kinematics of 3R planar manipulator respectively, and evaluate it by comparing the result of each function. ### **II. METHODS** #### **A. Forward Kinematics** The robot forward kinematics calculates the end-effector position and orientation given the joint angles and link lengths. The end-effector position is define by ![image.png](https://images.hive.blog/DQmePVaWstvAnKkXkDQKKuqipE6psNQJgDBthDeqJF1ZnLF/image.png) where θ comprises the joint angles and L is made up of all the link lengths. Also, ![image.png](https://images.hive.blog/DQmTuYN67A6ENaiEC2qZaQpHoEYNeM1xA2zPzRhzoptBfsq/image.png) We derived the joint angles using the trigonometric relations in a right triangle. The joint positions are ![image.png](https://images.hive.blog/DQmX46UUXyiNWjF3a7nAKYswxENX3VecNkBJx8B1UrLp8wB/image.png) We take the vector sum of all the joint position and yield the forward kinematic equation for 3R planar manipulator as ![image.png](https://images.hive.blog/DQmRJpMxW8Wwsu84dTag5wEj4P2T2pduASfYFJX1ZXFn1JH/image.png) #### **B. Reverse Kinematics** ![image.png](https://images.hive.blog/DQmbXsR7GaKG6W2jeaJv4LzhNX5SgVNBSP868Jw6RKj4RG7/image.png) Robot reverse kinematics is a more useful study because the end-effector position is known. In reverse kinematics problem, the task is to find the joint angles given position *p*, orientation *γ*, and the link lengths. Initially, we solve the position of *P3* and get ![image.png](https://images.hive.blog/DQmTZKu5N8bDqZXQQWc8r6RwKhPd9b9U57k5Te9cx7eBmnR/image.png) Hence *P3* is obtained by equation 7, the joint angles *θ*₁ and *θ*₂ are given by a 2R inverse kinematics problem with *P3* as the end-effector, as shown in Figure 3. We solve the angles *α* and *β* as ![image.png](https://images.hive.blog/DQmazNcSu8q7Uh4sQg7zCgsSYvCBVA2b5YDZZPm4HPNohm5/image.png) These angles are used to determine the joint angles *θ*₁ and *θ*₂. We keep in mind that there are two set of solution for the joint angles in 2R inverse kinematics problem. We refer joint 3 as the wrist, and joint 2 as the elbow. We compute the joint angles *θ*₁ and *θ*₂ by considering the elbow-up and elbow-down configuration, as shown in Figure 2, in reference to wrist angle *θ*₃. ![image.png](https://images.hive.blog/DQmXmZNjSWDzJ3J5xsRcHaBMBiZpKXoZYsADwMPAnq2P7Vh/image.png) The joint angles *θ*₁ and *θ*₂ are ![image.png](https://images.hive.blog/DQmQnPfWrZxiK3zHP78oo76NeAmpKDY6Zw7do7odAkaCgzH/image.png) The wrist angles, *θ*₃, are ![image.png](https://images.hive.blog/DQmUfWtncRniAXjFkYckgc5FjpWhJP9nLunLpwiqkx11EAP/image.png) The a and b notation denotes the elbow-up and elbow-down configuration respectively. The joint and wrist angles are shown in Figure 3. #### **C. MATLAB Implementation** We implement the equations for the forward and inverse kinematics of a 3R planar manipulator in MATLAB. We create a *fkinematics* and *ikinematics* function for forward and inverse kinematics respectively. The *fkinematics* function accept the link lengths and the joint angles. It returns the end-effector position and orientation. Also, a plot of the 3R planar manipulator and its end-effector point is displayed. The script used for *fkinematics* is shown in listing 1. ``` function fkinematics(links1,links2,links3,theta1,theta2,theta3) % Forward kinematics for 3-Link Planar Manipulator % Inputs: % links1,links2,links3 are length of each links in the robotic arm % base is the length of base of the robotic arm % theta1,theta2,theta3 are joint angles in reference to x-axis format compact format short L12 = links1; L23 = links2; L34 = links3; J1 = theta1; J2 = theta2; J3 = theta3; %joint equation x2 = L12*cosd(J1); x3 = L23*cosd(J1+J2)+ x2; xe = L34*cosd(J1+J2+J3) + x3; y2 = L12*sind(J1); y3 = L23*sind(J1+J2)+ y2; ye = L34*sind(J1+J2+J3) + y3; gamma = J1+J2+J3; fprintf('The position of the end-effector is (%f, %f) and orientation is(%f)\n',xe,ye,gamma) %plotting the links r = L12 + L23 + L34; daspect([1,1,1]) rectangle('Position',[-r,-r,2*r,2*r],'Curvature',[1,1],... 'LineStyle',':') hold on axis([-r r -r r]) line([0 x2], [0 y2]) line([x2 x3],[y2 y3]) line([x3 xe],[y3 ye]) line([0 0], [-r/10 r/10], 'Color', 'r') line([-r/10 r/10], [0 0], 'Color', 'r') hold on plot([0 x2 x3],[0 y2 y3],'o') plot([xe],[ye],'o','Color','r') grid on xlabel('x-axis') ylabel('y-axis') title('Forward Kinematics 3-Link Planar Manipulator') end ``` Listing 1: Forward Kinematics implementation (fkinematics function) We set the link lengths, the end-effector position, and orientation as the input for the *ikinematics* function. The function returns the joint angles and a plot showing the 3R planar manipulator. Hence the inverse cosine yields to either a real or complex value, we set a conditional statement in the script so that inverse cosine would return a real-valued angle. The script used for *ikinematics* is shown in listing 2. ``` function ikinematics(links1, links2, links3, positionx, positiony, gamma) % Inverse kinematics for 3-Link Planar Manipulator % Inputs: % links1,links2,links3 are length of each links in the robotic arm % base is the length of base of the robotic arm % positionx,positiony are joint angles in reference to x-axis % gamma is the orientation format compact format short L12 = links1; L23 = links2; L34 = links3; xe = positionx; ye = positiony; g = gamma; %position P3 x3 = xe-(L34*cosd(g)); y3 = ye-(L34*sind(g)); C = sqrt(x3^2 + y3^2); if (L12+L23) > C %angle a and B a = acosd((L12^2 + L23^2 - C^2 )/(2*L12*L23)); B = acosd((L12^2 + C^2 - L23^2 )/(2*L12*C)); %joint angles elbow-down J1a = atan2d(y3,x3)-B; J2a = 180-a; J3a = g - J1a -J2a; %joint angles elbow-up J1b = atan2d(y3,x3)+B; J2b = -(180-a); J3b = g - J1b - J2b; fprintf('The joint 1, 2 and 3 angles are (%f,%f, %f) respectively for elbow-down configuration.\n',J1a,J2a,J3a) fprintf('The joint 1, 2 and 3 angles are (%f,%f, %f) respectively for elbow-up configuration.\n',J1b,J2b,J3b) else disp(' Dimension error!') disp(' End-effecter is outside the workspace.') return end x2a = L12*cosd(J1a); y2a = L12*sind(J1a); x2b = L12*cosd(J1b); y2b = L12*sind(J1b); r = L12 + L23 + L34; daspect([1,1,1]) rectangle('Position',[-r,-r,2*r,2*r],'Curvature',[1,1],... 'LineStyle',':') line([0 x2a], [0 y2a],'Color','b') line([x2a x3], [y2a y3],'Color','b') line([x3 xe], [y3 ye],'Color','b') line([0 x2b], [0 y2b],'Color','g','LineStyle','--') line([x2b x3], [y2b y3],'Color','g','LineStyle','--') line([x3 xe], [y3 ye],'Color','b','LineStyle','--') %line([0 xe], [0 ye],'Color','r') line([0 0], [-r/10 r/10], 'Color', 'r') line([-r/10 r/10], [0 0], 'Color', 'r') hold on plot([0 x2a x3],[0 y2a y3],'o','Color','b') plot([x2b],[y2b],'o','Color','g') plot([xe],[ye],'o', 'Color', 'r') grid on xlabel('x-axis') ylabel('y-axis') title('Inverse Kinematics 3-Links Planar Manipulator') end ``` Listing 2: Inverse Kinematics implementation (ikinematics function) ### **III. Result and Discussion** ![image.png](https://images.hive.blog/DQmd7eRzMSBwP3VgE6XpMv8LSg889N8VnSbDNYSQoRRrFDT/image.png) #### *A. Forward Kinematics* In this section, we evaluate the forward and inverse kinematics equations of a 3R planar manipulator. We substitute the values shown in Figure 4 in the fkinematics. In the Matlab window console, we run ``` fkinematics(10,10,10,30,20,20) ``` and yield to the end-effector position, *(x*_e *= 18.508332, y*_e *= 22.057371)* and orientation, *(γ = 70.00)*. Figure 5 shows the 3R planar manipulator. ![image.png](https://images.hive.blog/DQmeBvEnBq5ysNzXXisibELBK6oJukhm84brZaM2RWHPpP4/image.png) The end-effector position and its orientation from the *fkinematics* is set as an input to *ikinematics* to evaluate it. The function must yield the joint angles shown in Figure 4. We run ``` ikinematics(10,10,10,18.508332,22.057371,70) ``` and we yield the following joint angles for elbow-down configuration are *θ*₁ *= 30.000009*, *θ*₂ *= 19.999981*, and *θ*₃ *= 20.000009*. Hence the inverse kinematics has two set of unique solution, we have the joint angles for elbow-up configuration as *θ*₁ *= 49.999991*, *θ*₂ *= −19.999981*, and *θ*₃ *= 39.999991*. ![image.png](https://images.hive.blog/DQmYcVikCHF5Zbba6NBuaeqMGLaJeyuBct31bsCsaZNoLb7/image.png) Thus, we verified that the forward kinematics equations which yields to a correct end-effector position and its orientation. ![image.png](https://images.hive.blog/DQmf4nM6G5eM1JTGHDs87V93D7uk2ccsypyRpCkvMpFSHRW/image.png) #### **B. Inverse Kinematics** Figure 7 shows a 3R planar manipulator with an identified end-effector position and its orientation. We use *ikinematics* function in MATLAB to generate the joint angles. Then, we evaluate the result by substituting it as an input to *fkinematics* function. In the Matlab window console, we run ``` ikinematics(10,10,10,15,20,85) ``` and yield a joint angle for elbow-down configuration as *θ*₁ *= 5.455370, θ* ₂ *= 59.875741, θ*₃ *= 19.668889* while for elbow-up configuration as *θ*₁ *= 65.331111, θ*₂ *= −59.875741, θ*₃ *= 79.544630*. Figure 9 shows the 3R planar manipulator plot for the end-effector position *x*_e *= 15*, *y*_e *= 20* and its orientation, *γ = 85*. ![image.png](https://images.hive.blog/DQmRtmgkXBXXAydACjsFzWdBKA8PBviSgzksk3mUSDejYaK/image.png) We validate the result by substituting the values to *fkinematics* and in Matlab console, run ``` fkinematics(10,10,10,5.455,59.876,19.669) ``` This gives us an end-effector position equal to *x*_e *= 15.000024* and *y*_e *= 19.999928*, and orientation equal to *γ = 85*. The joint angles in elbow-up configuration also yield to the same end-effector and orientation. ![image.png](https://images.hive.blog/DQmeh2yG2FEiCTv5jvyHn9xFfHBTLk2oNUnM76y2TS1xe6q/image.png) ![image.png](https://images.hive.blog/DQmc7rAAMd9H4mgHCjok1SvipzDBCehkJGfBTJ8pwgpKWyD/image.png) Thus, we verified that the inverse kinematics equations which yields to the joint angles. ### **IV. Conclusion** In this text, we modelled the equations for the forward and inverse kinematics of a 3R planar manipulator as shown in Figure 1. In MATLAB, we wrote a script for the *fkinematics* and *fkinematics* functions. These functions are used to validate the forward and invers kinematics model in the text. The *fkinematics* function returns the end-effector position *P* and its orientation *γ*. The result of fkinematics is validate by using its result as an input to ikinematics function. On the contrary, the ikinematics function returns the join angles θ₁, θ₂ and θ₃. We cross-validate the joint angles by setting it as an input to the fkinematic function. Thus, the forward and inverse kinematics model yield to correct values for the end-effector position and its orientation, and the joint angles. Also, the joint angles, either elbow-up and elbow-down configuration, in the inverse kinematics model gives the same end-effector point and orientation when cross-validated. ### **V. References** [1] [Lynch, K. and F. Park. “Modern Robotics: Mechanics, Planning, and Control.” (2017).](https://www.semanticscholar.org/paper/Modern-Robotics%3A-Mechanics%2C-Planning%2C-and-Control-Lynch-Park/ad094b63ac4c75239a5ea4464ecc79e73efd2fcd) [2] [John J. Craig. 1989. Introduction to Robotics: Mechanics and Control (2nd. ed.). Addison-Wesley Longman Publishing Co., Inc., USA.](http://www.mech.sharif.ir/c/document_library/get_file?uuid=5a4bb247-1430-4e46-942c-d692dead831f&groupId=14040) [3] [Merat, Frank. (1987). Introduction to robotics: Mechanics and control. Robotics and Automation, IEEE Journal of. 3. 166 - 166. 10.1109/JRA.1987.1087086](https://www.researchgate.net/publication/224729839_Introduction_to_robotics_Mechanics_and_control) (Note: All images in the text are personally drawn by the author except those with citations. ) Posted with [STEMGeeks](https://stemgeeks.net) Posted with [STEMGeeks](https://stemgeeks.net)

RE: INVESTMENT CHOICE ANNOUNCEMENT (AND A BRIEF TALK ON PALNET.IO)