19AIE201
Introduction to Robotics
IRB 140
19AIE201
Introduction to Robotics
Aadharsh Aadhithya - CB.EN.U4AIE20001
Anirudh Edpuganti - CB.EN.U4AIE20005
Madhav Kishor - CB.EN.U4AIE20033
Onteddu Chaitanya Reddy - CB.EN.U4AIE20045
Pillalamarri Akshaya - CB.EN.U4AIE20049
Team-1
IRB 140
Text
" Small, powerful, and fast 6-axes robot. Compact and very powerful. The IRB 140 six axes multipurpose industrial robot handles a payload of 6 kg with a reach of 810 mm (to axis 5) It can be floor mounted, inverted or wall mounted at any angle. The robust design with fully integrated cables adds to the overall flexibility and the collision detection function ensures the robot is reliable and safe."
\text{Rotation about } X_{i-1} \rightarrow \alpha_{i-1}
Link Twist
Z
Link Length
Z
\text{Translation along } X_{i-1} \rightarrow a_{i-1}
Joint Angle
X
\text{Rotation about } Z_{i} \rightarrow \theta_{i}
Link offset
X
\text{Translation along } Z_{i} \rightarrow d_{i}
\text{Rotation about } X_{i-1} \rightarrow \alpha_{i-1}
\text{Rotation about } Z_{i} \rightarrow \theta_{i}
\text{Translation along } Z_{i} \rightarrow d_{i}
\text{Translation along } X_{i-1} \rightarrow a_{i-1}
MDH Parameters
| 0 | 0 | 0.352 | |
| -90 | 0.070 | 0 | |
| 0 | 0.360 | 0 | |
| -90 | 0 | 0.380 | |
| 90 | 0 | 0 | |
| -90 | 0 | 0 |
\alpha_{i-1}
a_{i-1}
d_{i}
\theta_{i}
\theta_{1}
\theta_{2}-90
\theta_{3}
\theta_{4}
\theta_{5}
\theta_{5}
\theta_{6}
Forward Kinematics
\text{Joint angles} \rightarrow \text{End effector position}
T^0_6 = Tmdh(0,1)\cdot Tmdh(1,2)\cdot Tmdh(2,3)\cdot Tmdh(3,4)\cdot Tmdh(4,5)\cdot Tmdh(5,6)
MDH Parameters - Given
T^0_6 \rightarrow F(\theta_1,...,\theta_6)
MDH Parameters - Given
T^0_6 \rightarrow F(\theta_1,...,\theta_6)
{\theta_1,...,\theta_6} \rightarrow Input
T^0_6 \rightarrow F(\theta_1,...,\theta_6)
{\theta_1,...,\theta_6} \rightarrow Input
Inverse Kinematics
\text{End effector position} \rightarrow \text{Joint angles}
T^0_6 \rightarrow Given
MDH Parameters - Given
T^0_6 \rightarrow Given
MDH Parameters - Given
t^0_6 \rightarrow obtained
t^0_6 =T^0_6
t^0_6 =T^0_6
How to solve this??
Before that, lets see an application based on IK
Before that, lets see an application based on IK
Pantograph
19AIE201
Introduction to Robotics
Closed form solution
Closed Form Solution
\theta_1 = atan2(y,x)
\theta_1 = \pi + atan2(y,x)
Closed Form Solution
Closed Form Solution
cos(\zeta) = \frac{s^2 + r^2 - a_2^2 - (d_4 + d_6)^2}{2 a_2 (d_4 + d_6)}
sin(\zeta) = \pm \sqrt{1 - cos^2(\zeta)}
\zeta = atan2(sin(\zeta) , cos(\zeta))
\theta_3 = -(\frac{\pi}{2} + \zeta)
Closed Form Solution
\theta_3 = -(\frac{\pi}{2} + \zeta)
\theta_2 = atan2(s,r) - atan2((d_4 + d_6)sin(\zeta) , a_2 + (d_4 + d_6)cos(\zeta))
\theta_3 = -(\frac{\pi}{2} + \zeta)
\theta_2 = atan2(s,r) - atan2((d_4 + d_6)sin(\zeta) , a_2 + (d_4 + d_6)cos(\zeta))
\theta_1 = atan2(y,x)
\theta_4, \theta_5 , \theta_6 \text{ can be found by comparing to}\\
R^3_6
19AIE201
Introduction to Robotics
Newton-Raphson method
Newton-Raphson method
^0_6T_{\theta} = \ ^0_6T(\theta_1,\theta_2,\theta_3,\theta_4,\theta_5,\theta_6)
^0_6T_{des} \ is \ a \ final \ orientation \ matrix \ of \ end \ effector \\
Let \ ^0_6T_{\theta}(a,b) \ denote \ the \ element \ of \ matrix \ ^0_6T_{\theta} \ that \ is \ in \ the \ ath \ row \ and \ bth \ column
\theta = \ ^0_6T_{\theta} - ^0_6T_{des}
Newton-Raphson method
\theta_{n+1} = \theta_n - \frac{F(\theta_n)}{F'(\theta_n)}
Newton-Raphson method
\theta_{n+1} = \theta_n - \frac{F(\theta_n)}{F'(\theta_n)}
\theta_{n+1} = \theta_n - J(\theta_n)^{-1}F(\theta_n)
Newton-Raphson method
\theta_{n+1} = \theta_n - J(\theta_n)^{-1}F(\theta_n)
Jacobian Matrix
19AIE201
Introduction to Robotics
Trajectory Planning
Trajectory Planning
Tracking the change in position, velocity and acceleration of joints w.r.t the time.
Trajectory Planning
Trajectory Planning
Polynomial Function
Trajectory Planning
Polynomial Function
\theta(t) = a_0 + a_1t + a_2t^2 + a_3t^3
Trajectory Planning
Polynomial Function
\theta(t) = a_0 + a_1t + a_2t^2 + a_3t^3
\dot{\theta(t)} = a_1 + 2a_2t + 3a_3t^2
Trajectory Planning
Polynomial Function
\theta(t) = a_0 + a_1t + a_2t^2 + a_3t^3
\dot{\theta(t)} = a_1 + 2a_2t + 3a_3t^2
\ddot{\theta(t)} = 2a_2 + 6a_3t
Trajectory Planning
Constraints
Trajectory Planning
Constraints
\theta(0) = \theta_0
Trajectory Planning
Constraints
\theta(t_f) = \theta_f
\theta(0) = \theta_0
Trajectory Planning
Constraints
\theta(t_f) = \theta_f
\theta(0) = \theta_0
\dot{\theta(0)} = 0
Trajectory Planning
Constraints
\theta(t_f) = \theta_f
\theta(0) = \theta_0
\dot{\theta(0)} = 0
\dot{\theta(t_f)} = 0
Trajectory Planning
Using these constraints...
\theta(t_f) = \theta_f
\theta(0) = \theta_0
\dot{\theta(0)} = 0
\dot{\theta(t_f)} = 0
\theta(t) = a_0 + a_1t + a_2t^2 + a_3t^3
\dot{\theta(t)} = a_1 + 2a_2t + 3a_3t^2
\ddot{\theta(t)} = 2a_2 + 6a_3t
Trajectory Planning
Using these constraints...
\theta(t_f) = \theta_f
\theta(0) = \theta_0
\dot{\theta(0)} = 0
\dot{\theta(t_f)} = 0
\theta(t) = a_0 + a_1t + a_2t^2 + a_3t^3
\dot{\theta(t)} = a_1 + 2a_2t + 3a_3t^2
\ddot{\theta(t)} = 2a_2 + 6a_3t
\Rightarrow
Trajectory Planning
In these equations...
\theta(t_f) = \theta_f
\theta(0) = \theta_0
\dot{\theta(0)} = 0
\dot{\theta(t_f)} = 0
\theta(t) = a_0 + a_1t + a_2t^2 + a_3t^3
\dot{\theta(t)} = a_1 + 2a_2t + 3a_3t^2
\ddot{\theta(t)} = 2a_2 + 6a_3t
\Rightarrow
Trajectory Planning
We get this
a_0 = \theta_0
a_1 = 0
a_2 = \frac{3}{t^2}(\theta_f - \theta_0)
a_3 = -\frac{2}{t^3}(\theta_f - \theta_0)
19AIE201
Introduction to Robotics
Jacobian
Jacobian
x = f_1(\theta_1 ,\theta_2 ,\theta_3 ,\theta_4 ,\theta_5 ,\theta_6 )\\
y = f_2(\theta_1 ,\theta_2 ,\theta_3 ,\theta_4 ,\theta_5 ,\theta_6 )\\
z = f_3(\theta_1 ,\theta_2 ,\theta_3 ,\theta_4 ,\theta_5 ,\theta_6 )
J_v
Jacobian
\omega_i = \dot{q_i}\hat{z_i}
J_{\omega}
Jacobian
J = \begin{bmatrix}
J_v & J_{\omega}
\end{bmatrix}
\begin{bmatrix}
v_x \\
v_y\\
v_z\\
\omega_x\\
\omega_y\\
\omega_z
\end{bmatrix} = J\cdot \vec{\dot{q}}
Thank you sir
ROBOTICS
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