1. Dynamics and Control of Hexapod Systems
Jass 2006
St. Petersburg
Dynamics and Control of Hexapod Systems

2. Table of Contents What is a Hexapod? History
Advantages and disadvantages of a Hexapod
Examples of application
Mathematical description – Kinematics
Mathematical descriptions – Dynamics
Controlling of a Hexapod
Simulation of the controller
JASS 2006 – St. Petersburg
Daniela Gewald

3. What is a Hexapod?
greek language: “hexa” = 6, “podus” = foot, also called Stewart-Gough-Platform
a hexapod belongs to the group of parallel kinematics
in general it consists of:
by varying the length of the axes, motions in all six degrees of freedom can be generated
upper platform
joints
six parallel axes, variable in length
lower platform
JASS 2006 – St. Petersburg
Daniela Gewald

4. Movements of a Hexapod 1 JASS 2006 – St. Petersburg Daniela Gewald
JASS 2006 – St. Petersburg
Daniela Gewald

5. Movements of a Hexapod 2 JASS 2006 – St. Petersburg Daniela Gewald
JASS 2006 – St. Petersburg
Daniela Gewald

6. A little bit of History
1955: Invention of the hexapod; V. Gough was the first, who realized such type of robot for testing aircraft tyres under six component loading
1965: D. Stewart presented a parallel mechnism to be used as motion simulator
1999 – 2004: invention and testing of a hexapod telescope by Krupp and the Ruhr University of Bochum; now this telescope is working in Antofagasta (Chile)
JASS 2006 – St. Petersburg
Daniela Gewald

7. Advantages and Disadvantages
high system stiffness as a result of the parallel system structure
high load/weight ratio
non-cumulative joint errors
lot of similar parts  reduction of producing costs
precise movements, also under heavily alternating loads and high accelerations
arrangement of all motors for the single axes near the basis  strong basis motor is not necessary
Main disadvantage:
strictly limited workspace, determined by the axes stroke and the maximum angle of the joints
JASS 2006 – St. Petersburg
Daniela Gewald

8. Examples of Hexapod systems 1
motion generator for flight simulators
applications in medical technology
JASS 2006 – St. Petersburg
Daniela Gewald

9. Examples of Hexapod systems 2
antenna precision alignment; motion control for telescopes
applications in machine tools for highly precise motions between tool and work piece
JASS 2006 – St. Petersburg
Daniela Gewald

10. Mathematical description of a Hexapod 1
Kinematics of the hexapod system:
geometrical relations at one leg
TCP
upper platform
upper joint
upper leg
lower leg
lower joint
lower platform
JASS 2006 – St. Petersburg
Daniela Gewald

11. Mathematical description of a Hexapod 2
inverse kinematics: calculating with a given TCP vector
direct kinematics: calculating the position of the TCP, when the leg length is givenproblem can be solved numerically with the Newton-approach-scheme
JASS 2006 – St. Petersburg
Daniela Gewald

12. Mathematical description of a Hexapod 3
calculation of the lower joint angles:
for the calculation two more are introduced:
pre-orientated by the constant z-angle γ0  resulting from the mounting angle of the universal joint
this frame is fixed to each leg and rotates with the universal joints
JASS 2006 – St. Petersburg
Daniela Gewald

13. Mathematical description of a Hexapod 4
lower joint angles:
JASS 2006 – St. Petersburg
Daniela Gewald

14. Mathematical description of a Hexapod 5
upper joint angles:
similar way with a new frame for the upper joints
(constant pre-orientation γ1)
transformation of in the local frame :
JASS 2006 – St. Petersburg
Daniela Gewald

15. Mathematical description of a Hexapod 6
cardan errors:
in a system with the combination of two universal joints
and a screw joint a yet unconsidered degree of freedom
(DOF) occurs: a passive rotation of the upper part of
the leg around the axis of the corresponding lower part
this rotation is given by:
JASS 2006 – St. Petersburg
Daniela Gewald

16. Mathematical description of a Hexapod 7
Platform Jacobian:
the Jacobian is describing the relationship between the
norm of the leg velocities and the TCP velocities
by defining an unit vector:
we can write the norm of the leg velocities:
simplifying the equation by using the “skew symmetric tilde
tensor”:
JASS 2006 – St. Petersburg
Daniela Gewald

17. Mathematical description of a Hexapod 8
Platform Jacobian:
JASS 2006 – St. Petersburg
Daniela Gewald

18. Mathematical description of a Hexapod 9
Singularities of the Jacobian:
important for the stability of the model: has to be
nonsingular
possible singularities are:
parameterization of the rotation matrix TIM: if one of the joint angles is at 90° a sinularity may appear
Jacobian is not regular: structural singularity due to infinite leg forces
the Jacobian will also be singular, if the hexapod looses one degree of freedom
to avoid this three cases, joint limits of the hexapod system will be exceeded befor a singularity can occur
determinant of can be used as a measure for the model configuration
JASS 2006 – St. Petersburg
Daniela Gewald

19. Dynamics of a Hexapod 1
Notation of the Newton-Euler equations in connection with
Jacobians:
as mentioned before: Jacobians describe the relation of
velocities between independent coordinates and
the coordinates of the corresponding body
velocities:
accelerations:
JASS 2006 – St. Petersburg
Daniela Gewald

20. Dynamics of the Hexapod 2
if the complete state of each body is known, the dynamics
of the hexapod can be derives in closed form by means of
the Newton-Euler-Matrix-formulation:
with:
the active forces:
change of linear momentum:
change of angular momentum:
JASS 2006 – St. Petersburg
Daniela Gewald

21. Dynamics of the Hexapod 3
Dynamics of the upper platform:
JASS 2006 – St. Petersburg
Daniela Gewald

22. Dynamics of the Hexapod 4
Dynamics of the upper platform:
vector can be changed with respect to the center of gravity of another body, wich is fixed on the upper platform
active forces: just gravity forces
independent TCP coordinates = platform degrees of freedom
JASS 2006 – St. Petersburg
Daniela Gewald

23. Equations of Motion J= Jacobi Matrix M= Mass Matrix
G= Matrix of centrifugal and gyroscopic forces
g= Matrix for gravitation forces
JASS 2006 – St. Petersburg
Daniela Gewald

24. Control of a hexapod system
The controlling task can be divided in 2 parts:
Tracking or positioning task
Isolation of the upper platform from ground vibrations
JASS 2006 – St. Petersburg
Daniela Gewald

25. Controlling of an one – DOF – System 1
 Ideal stabilization at an example of a simple, rigid and friction free one degree of freedom system
The undesired vibrations (z(t)) affecting the base platform
are transferred to the upper platform, thus the servo
moment has to hold the platform statically and keep q at a
desired constant value
JASS 2006 – St. Petersburg
Daniela Gewald

26. Controlling of an one – DOF – System 2
Feedforward control for the ideal stabilization +
One-DOF-system +
Perfect vibration isolation, if
with =pitch of the spindle
JASS 2006 – St. Petersburg
Daniela Gewald

27. Controlling of a Hexapod – System 1
Hexapod at the Department of Applied Mechanics (TUM)
Topology of one hexapod leg
JASS 2006 – St. Petersburg
Daniela Gewald

28. Controlling of a Hexapod – System 2
Differences between the hexapod and the one-DOF-
system:
more complex equations, because of closed kinematical
loops  this loops are the reason for the high stiffness of
the system, which allows to assume the hexapod as a rigid
Multi-Body-System (MBS)
Effects to be considered, when designing a controller:
parameter uncertainties
external disturbance forces
measurement noise
undesired TCP motions may occur  besides the feedforward controller a feedback controller, that can compensate residual vibrations of the TCP, is needed
also important: both tasks (tracking control and vibration isolation) should be treated by the controller
JASS 2006 – St. Petersburg
Daniela Gewald

29. Controlling of a Hexapod – System 3
Assumptions:
q=constant
vibrations z(t) affect the
lower platform
Servo Moment:
J= Jacobi Matrix
M= Mass Matrix
G= Matrix of centrifugal and gyroscopic forces
g= Matrix for gravitation forces
JASS 2006 – St. Petersburg
Daniela Gewald

30. Controlling of a Hexapod – System 4
Servo Forces:
in the case of ideal vibration isolation we need finally a
force control that is sensitive to the effects we listed before
equations of motion, when the TCP is moving, and no
vibrations (z=0) are present:
the hexapod dynamics are decoupled, if the servo forces
are chosen as:
linear dynamics of the hexapod
six independent double integrators
JASS 2006 – St. Petersburg
Daniela Gewald

31. Controlling of a Hexapod – System 5
Design of the controller with the PD-law:
two control parameters KP and KD
constraint for the parameters: asymptotic behaviour of the system = no overshoots
complete control law for the hexapod:
and when we take into account the undesired vibrations z(t):
JASS 2006 – St. Petersburg
Daniela Gewald

32. Controlling of a Hexapod – System 6
block diagram of the stabilization/tracking controller
JASS 2006 – St. Petersburg
Daniela Gewald

33. Simulation of the controlled System 1
just isolation of vibrations excited by noise (0-5Hz) around the two horizontal axes (x, y)
JASS 2006 – St. Petersburg
Daniela Gewald

34. Simulation of the controlled System 2
before 5s: controller is inactive
after 5s: high isolation performance
 because of the coupling of the hexapod-dynamics, the controller excites vibrations around the vertical axis (z, γ), but with relatively small amplitudes
JASS 2006 – St. Petersburg
Daniela Gewald

35. Simulation of the controlled System 3
platform stabilization and tracking control (circle path in the x-y-plane, r=120mm)
 again very high performance: maximum path deviation in the x-y-plane < 1,4mm, maximum deviation of rotation < 0,18°, deviation in vertical direction < 0,2mm
JASS 2006 – St. Petersburg
Daniela Gewald

36. Simulation of the controlled System 4
JASS 2006 – St. Petersburg
Daniela Gewald

37. Simulation of the controlled System 5
JASS 2006 – St. Petersburg
Daniela Gewald