Hugo Hadfield
Cambridge University PhD student, Signal Processing and Communications Laboratory
The Forward and Inverse Kinematics of a Delta Robot
Hugo Hadfield, Lai Wei, Joan Lasenby
University Of Cambridge,
Signal Processing and Communications Laboratory
Speaker: Hugo Hadfield
Geometry of a Delta Robot
Image source: AEMK Systems
Inverse Kinematics
Inverse Kinematics
Forward Kinematics
Forward Kinematics
Derivatives for the Jacobians
- In order to calculate the jacobians we will need some derivatives, let's put together a little collection:
- Derivative of a CGA point with respect to a parameter of the 3D point
- Derivative of the intersection of a circle and a sphere with respect to a parameter of the sphere centre
- Derivative of the intersection of three spheres to make a point pair
- Derivative of the extraction of the end points from a point pair
- Derivative of a 3D point with respect to parameters of its conformal equivalent
Derivative of a CGA point with respect to a parameter of the 3D point
- The 3D point \(x_i\) can then be represented as the CGA point \(X_i\):
X_i = \frac{1}{2}x_i^2n_\infty + x_i + n_0
\frac{\partial X_i}{\partial \alpha} = \left( \frac{\partial x_i}{ \partial \alpha}\cdot x_i \right)n_\infty + \frac{\partial x_i}{\partial \alpha}
- We can then take the derivative of the CGA point with respect to a parameter of the 3DGA point, \(\alpha\) :
Derivative of the intersection of a circle and a sphere with respect to a parameter of the sphere centre
- Consider the form of the dual CGA sphere:
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\pdiff{T}{\alpha} = \left(\pdiff{\Sigma^*}{\alpha}\wedge C^*\right)^*
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\Sigma^* = X - \frac{1}{2}\rho^2\ninf
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\pdiff{\Sigma^*}{\alpha} = \pdiff{X}{\alpha}
- Take derivatives with respect to a parameter \(\alpha\) of the centre of the sphere:
- Finally consider the meet of the sphere and a fixed circle C, giving a point pair T:
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T = \left(\Sigma^* \wedge C^* \right)^*
Derivative of the intersection of three spheres to make a point pair
- The point pair at the intersection of three spheres is given by:
T = (\Sigma_1\vee \Sigma_2 \vee \Sigma_3) \equiv (\Sigma_1^*\wedge \Sigma_2^*\wedge \Sigma_3^*)^*
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\pdiff{T}{\theta_1} = \left(\pdiff{\Sigma_1^*}{\theta_1}\wedge \Sigma_2^*\wedge \Sigma_3^*\right)^*
, \hspace{0.1cm}
\pdiff{T}{\theta_2} = \left( \Sigma_1^*\wedge \pdiff{\Sigma_2^*}{\theta_2}\wedge \Sigma_3^*\right)^*
\\
\pdiff{T}{\theta_3} = \left(\Sigma_1^* \wedge \Sigma_2^*\wedge \pdiff{\Sigma_3^*}{\theta_3}\right)^*
- And so the derivatives of the point pair with respect to a parameter of each sphere individually is:
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\pdiff{T}{\theta_i} = (-1)^{i-1}\left(\pdiff{\Sigma_i^*}{\theta_i}\wedge C^*\right)^*
, \hspace{0.1cm} \text{where} \,\,\, C^* = \bigwedge_{j \in {1,2,3} \,\,\, j\ne i}\Sigma_j^*
- This can also be written as:
Derivative of the extraction of the end points from a point pair
- To extract the end points of a point pair T we will follow the method given in [CovApproach]. First construct a projector:
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P = \frac{1}{2}\left(1 + \frac{T}{\sqrt{T^2}}\right)
, \hspace{0.2cm}
\pdiff{P}{\alpha} = \frac{1}{2T^2}\left(\sqrt{T^2}\pdiff{T}{\alpha} - T\frac{\pdiff{T}{\alpha}\cdot T}{\sqrt{T^2}}\right)
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Y = -\tilde{P}(T\cdot n_\infty)P
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\pdiff{Y}{\alpha} = -\pdiff{\tilde{P}}{\alpha}(T\cdot n_\infty)P -\tilde{P}\left(\pdiff{T}{\alpha}\cdot n_\infty\right)P -\tilde{P}(T\cdot n_\infty)\pdiff{P}{\alpha}
- Then use it to project out a point:
Derivative of a 3D point with respect to parameters of its conformal equivalent
- To map back from a conformal point to a standard point we need to normalise by \(Y\cdot n_\infty\)
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y = \frac{-\sum_{j=1}^{j=3}(Y\cdot e_j)e_j}{Y\cdot n_\infty}
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\pdiff{y}{\theta_i} = \frac{ -\sum_{j=1}^{j=3}\left(\pdiff{Y}{\theta_i}\cdot e_{j}\right)e_j (Y\cdot\ninf) + \sum_{j=1}^{j=3}\left(Y\cdot e_{j}\right)e_j (\pdiff{Y}{\theta_i}\cdot\ninf) }{ (Y\cdot\ninf)^2 }
- And so we can calculate our final derivative:
Forward Kinematic Jacobian
Inverse Kinematic Jacobian Horizontal
Inverse Kinematic Jacobian Vertical
Verification and Visualisation of the Robot in Clifford Python
- We have implemented all of these derivatives in Clifford Python and tested them vs central differences with visualisation via pyganja
Designing Practical Controllers for the Delta Robot
- To design practical controllers for real robots we need to do some kind of dynamic modelling
- We also need a mechanism for modelling environmental disturbances
- We would like someone else to write the physics code - not the point of this paper
- We would like easy visualisation of everything in real time
Game engines tick all the boxes
Setting up Unity3D as a Simulation Environment
- Unity3D is programmed in C#
- Generated a C# library with ganja.js
- Fleshed out library to integrate with Unity3D
- Built a physical model of the robot
- Attached the CGA controller to it
Testing CGA based Controllers on Simulated Robots in Unity3D
Testing CGA based Controllers on Simulated Robots in Unity3D
Future Work
- Constructing a dynamic model of the delta robot within our GA framework will let us use more interesting controller designs and test physical design parameters
- We plan to connect our simulated robot and controller to a real Delta robot via ROS
- We plan to improve the Unity3D C# (C)GA library to allow fast iteration of robot/games/graphics related applied research
Conclusions
- (C)GA can provide a very useful framework for describing the kinematics of parallel robots
- Putting together a "data book" of derivatives of common operations in (C)GA would be very useful to practitioners trying to use GA in practice/in industry (boring but comprehensive is good?)
- There already exists a wide range of modern GA software packages of high quality but integration with game engines is very useful and hence desirable
Forward and Inverse Kinematics of the Delta Robot
By Hugo Hadfield
Forward and Inverse Kinematics of the Delta Robot
In this presentation we cover the geometry that underlies the forward and inverse kinematic equations of the Delta robot, we present the mathematics in Conformal Geometric Algebra and have interactive demos.
