kinematic modelling of continuum robots following constant...
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Kinematic Modelling of continuum robots following
constant curvature
Centre for Robotics Research – School of Natural and Mathematical Sciences – King’s College London
Hongbin Liu
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Compliant Robotics Peking University, Globex, July 20182
• Continuum robots are increasing popular in modern robotics
• Continuous design can be highly advantageous for
– Compliant adaptation to unknown environments
– Safe manipulation of fragile objects and in human-robot interaction
Introduction to Constant Curvature Model
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Compliant Robotics Peking University, Globex, July 20183
• Unlike rigid-linked, hyper-redundant robots no discrete links
• Continuous deformation of robot’s body allows for motions such as
– Bending around unknown objects for manipulation
– Dexterous movements for locomotion in uncertain terrain
Introduction to Constant Curvature Model
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Compliant Robotics Peking University, Globex, July 20184
• Until now:
– Robot fully defined for a given set of joint angles and link lengths
Introduction to Constant Curvature Model
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Compliant Robotics Peking University, Globex, July 20185
• Until now:
– Robot fully defined for a given set of joint angles and link lengths
• Now Continuum robots:
– Underactuated system; infinite dofs have to be addressed while only limited number of dofs can be controlled
– Forces and moments have to be considered due to inherent elastic behaviour
Introduction to Constant Curvature Model
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Compliant Robotics Peking University, Globex, July 20186
• Until now:
– Robot fully defined for a given set of joint angles and link lengths
• Now Continuum robots :
– Underactuation of the system; infinite dofs have to be addressed while only limited number of dofs can be controlled
– Forces and moments have to be considered due to inherent elastic behaviour
Assumptions necessary to recreate shape of robot
Introduction to Constant Curvature Model
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Compliant Robotics Peking University, Globex, July 20187
Definition of Constant Curvature
• Shape of robot can be recreated assuming a Constant Curvature
• The reason will be discussed in Mechanical Modelling
• CC allows for geometrical description of robot following a curve with curvature k, bending radius r, angle of bending plane φ and arch length l
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Compliant Robotics Peking University, Globex, July 20188
• Model can be generated based on mappings between three spaces
– Actuator (tendon driven) space
– Configuration space
– Task space
• Each space comprises set of descriptive variables and spaces changed using mapping functions
Mapping
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Compliant Robotics Peking University, Globex, July 20189
• Robot configuration can be expressed as a set of previously defined arc parameters (k, φ and l)
• Given these parameters the kinematic description can be achieved applying universal modelling techniques, independent from robot type
Mapping
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Compliant Robotics Peking University, Globex, July 201810
• Robot configuration can be expressed as a set of previously defined arc parameters (k, φ and l)
• Given these parameters the kinematic description can be achieved applying universal modelling techniques, independent from robot type
• Mapping from actuator to configuration space is robot-specific
Mapping
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Compliant Robotics Peking University, Globex, July 201811
Example Tendon driven:
Continuously bending actuators
• Tendon driven leading to length change
• Placement of actuators allows bending motions of robot
• Robot tip position and orientation can be described as a
function of the actuator lengths
Mapping: Actuator to Config. space
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Compliant Robotics Peking University, Globex, July 201812
Example: Tendon driven
CMU snake robot
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Compliant Robotics Peking University, Globex, July 201813
Example: Tendon driven
OC snake robot, UK
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Compliant Robotics Peking University, Globex, July 201814
Example: Tendon driven
Wire-Driven Flexible Robot Arm
The china university of Hong Kong
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Compliant Robotics Peking University, Globex, July 201815
Example: fluidic actuation:
Continuously bending actuators
• Inflatable chambers leading to length change upon pressurization
• Placement of actuators allows elongation and bending motions of robot
• Robot tip position and orientation can be described as a
function of the actuator lengths
Mapping: Actuator to Config. space
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Compliant Robotics Peking University, Globex, July 201816
Example: fluidic actuation
Festo arm, Germany
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Compliant Robotics Peking University, Globex, July 201817
Example: fluidic actuation
Stiff-Flop manipulatorEU FP7 King’s College London, UK
• Silicone-based soft body
• Multi-segment continuum robot with
3 dofs per element
• Fluidic actuation (air pressure or
hydraulic)
• Multi-purpose platform for minimally-
invasive surgery
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Compliant Robotics Peking University, Globex, July 201818
Example: fluidic actuation
L1L3L2
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Compliant Robotics Peking University, Globex, July 201819
Example: fluidic actuation
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Compliant Robotics Peking University, Globex, July 201820
Constant curvature in 2D
𝑙1 = 𝜃𝑟𝑙2 = 𝜃 𝑟 + 2𝑑𝑙 = 𝜃(𝑟 + 𝑑)
𝑙2 = 𝜃 𝑟 + 2𝑑 + 𝜃𝑟 − 𝜃𝑟
𝑙2 = 2𝜃 𝑟 + 𝑑 − 𝜃𝑟𝑙2 = 2𝑙 − 𝑙1
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼𝑙 =
𝑙2 + 𝑙12
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Compliant Robotics Peking University, Globex, July 201821
Constant curvature in 2D
𝑙2 − 𝑙1 = 2𝜃𝑑
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼
𝜃 =𝑙2 − 𝑙12𝑑
𝑟 =1
𝑘=
2𝑙1𝑑
𝑙2−𝑙1
𝑙1 = 𝜃𝑟𝑙2 = 𝜃 𝑟 + 2𝑑𝑙 = 𝜃(𝑟 + 𝑑)
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Compliant Robotics Peking University, Globex, July 201822
Constant curvature in 3D
bending plane
base plane
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Compliant Robotics Peking University, Globex, July 201823
Arc parameters: Length
• Given the bending geometry it can be seen that
Constant curvature in 3D
𝑑 ∙ cosΦi
𝑟
𝜃
𝑟𝑖
bending plane base plane
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Compliant Robotics Peking University, Globex, July 201824
Arc parameters: Length
• Substituting the arc lengths and this relation becomes
Constant curvature in 3D
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Compliant Robotics Peking University, Globex, July 201825
Arc parameters: Length
• The relations between and the bending planecan be denoted as
• Substituting these relations into yields the expression
Constant curvature in 3D
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Compliant Robotics Peking University, Globex, July 201826
Arc parameters: Bending angle
• To obtain the bending angle, the previously derived equation
is to be rearranged for actuators 1&2 and equated leading to
• This procedure can be repeated for all actuator pairs and rearranged, leading to the expression
Constant curvature in 3D
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Compliant Robotics Peking University, Globex, July 201827
Proof
sin ( α ± β ) = sinα cosβ ± cosα sinβ cos ( α ± β ) = cosα cosβ ∓ sinα sinβ
cos 𝜙1 = cos𝜋
2− 𝜙 = 𝑠𝑖𝑛𝜙
cos 𝜙3 = cos11𝜋
6− 𝜙 = 𝑐𝑜𝑠
11𝜋
6𝑐𝑜𝑠𝜙 + sin
11𝜋
6𝑠𝑖𝑛𝜙
3
2𝑐𝑜𝑠𝜙 −
1
2𝑠𝑖𝑛𝜙
cos 𝜙2 = cos7𝜋
6− 𝜙 = 𝑐𝑜𝑠
7𝜋
6𝑐𝑜𝑠𝜙 + sin
7𝜋
6𝑠𝑖𝑛𝜙
−3
2𝑐𝑜𝑠𝜙 −
1
2𝑠𝑖𝑛𝜙
Recall
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Compliant Robotics Peking University, Globex, July 201828
Proof
𝜃𝑑 =𝑙2 − 𝑙1
𝐶𝑂𝑆𝜙1 − 𝐶𝑂𝑆𝜙2
𝜃𝑑 =𝑙3 − 𝑙1
𝐶𝑂𝑆𝜙1 − 𝐶𝑂𝑆𝜙3
3
2𝑠𝑖𝑛𝜙 +
3
2𝑐𝑜𝑠𝜙 = (𝑙2 − 𝑙1)/𝜃𝑑
3
2𝑠𝑖𝑛𝜙 −
3
2𝑐𝑜𝑠𝜙 = (𝑙3 − 𝑙1)/𝜃𝑑
(1)
(2)
(1)+(2)
(1)-(2)
3𝑠𝑖𝑛𝜙 = (𝑙2 + 𝑙3 − 2𝑙1)/𝜃𝑑
3𝑐𝑜𝑠𝜙 = (𝑙2 − 𝑙3)/𝜃𝑑
𝑡𝑎𝑛𝜙 =3
3
(𝑙2 + 𝑙3 − 2𝑙1)
(𝑙2 − 𝑙3)
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Compliant Robotics Peking University, Globex, July 201829
Arc parameters: Curvature
Based on geometric observations it can be derived that
Substituting into and (r=1/K) yields
Consider actuator 1, 𝜙1 = 90 − 𝜙
𝜅 =𝑙2 + 𝑙3 − 2𝑙1
𝑙1 + 𝑙2 + 𝑙3 𝑑 c𝑜𝑠Φ1
Constant curvature in 3D
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Compliant Robotics Peking University, Globex, July 201830
Consider actuator 1, 𝜙1 = 90 − 𝜙
𝜅 =𝑙2 + 𝑙3 − 2𝑙1
𝑙1 + 𝑙2 + 𝑙3 𝑑 sin𝛷
Constant curvature in 3D
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Compliant Robotics Peking University, Globex, July 201831
Arc parameters: Curvature
Given the curvature derivation and previous expression for the bending with the identity
sin(tan−1𝑦
𝑥) = 𝑦/ 𝑥2 + 𝑦2
It can be seen that
Constant curvature in 3D
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Compliant Robotics Peking University, Globex, July 201832
Overview
Arc parameters of a 3dof continuum robot with actuator lengths 𝑙𝑖 can be represented as
Forward mapping
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Compliant Robotics Peking University, Globex, July 201833
Arc geometry
• Shape of the robot can be expressed applying the transformation
Where R represent rotation matrix and p is the translation vector
Mapping: Config to Task space-2D
𝐴10 =
𝑹 𝒑0 1
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Compliant Robotics Peking University, Globex, July 201834
Mapping: Config to Task space-2D
𝑙 =𝑙2 + 𝑙12
𝜃 =𝑙2 − 𝑙12𝑑
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼
𝛼 =𝜋
2−𝜃
2𝜃
2
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Compliant Robotics Peking University, Globex, July 201835
Mapping: Config to Task space-2D
𝑙 =𝑙2 + 𝑙12
𝜃 =𝑙2 − 𝑙12𝑑
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼
𝛼 =𝜋
2−𝜃
2
𝑜𝑜1 = 2𝑙
𝜃𝑐𝑜𝑠𝛼
𝑜1 =2𝑙
𝜃𝑐𝑜𝑠𝛼 2,
2𝑙
𝜃𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛼
𝑇
𝜃
2
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Compliant Robotics Peking University, Globex, July 201836
Mapping: Config to Task space-2D
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼
𝛼 =𝜋
2−𝜃
2
𝑜1 =2𝑙
𝜃𝑐𝑜𝑠𝛼 2,
2𝑙
𝜃𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛼
𝑇
𝑅 =cos(−𝜃) −sin(− 𝜃)sin(−𝜃) cos(−𝜃)
𝐴10 = 𝑅 𝑜1
0 1
𝑝01
= 𝐴10 𝑝1
1
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Compliant Robotics Peking University, Globex, July 201837
Mapping: Config to Task space-2D- Singularity
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼
𝛼 =𝜋
2−𝜃
2
𝑜1 =2𝑙
𝜃𝑐𝑜𝑠𝛼 2,
2𝑙
𝜃𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛼
𝑇
𝜃 =𝑙2 − 𝑙12𝑑
Singularity problem:
When θ is zero, o1 become undefined
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Compliant Robotics Peking University, Globex, July 201838
Programming exercise in class
change name “transcc2D_q.m” to “transcc2D.m”
complete this function and type in the commend window:
[A o1]=transcc2D(pi/4)
results:A = 0.7071 0.7071 3.7292
-0.7071 0.7071 9.0032 0 0 1.0000
o1 = 3.7292 9.0032
Mapping: Config to Task space-2D
![Page 39: Kinematic Modelling of continuum robots following constant ...globex.coe.pku.edu.cn/file/upload/201807/05/0759057018.pdf · Definition of Constant Curvature • Shape of robot can](https://reader033.vdocuments.us/reader033/viewer/2022060320/5f0d18e17e708231d438aa0b/html5/thumbnails/39.jpg)
Compliant Robotics Peking University, Globex, July 201839
Programming exercise in class
change name “transcc2D_q.m” to “transcc2D_L.m”
change this function to : [A o1]=transcc2D_L(L1, L2)
Input variables are the two tendon lengths L1 and L2
Mapping: Config to Task space-2D
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Compliant Robotics Peking University, Globex, July 201840
Mapping: Config to Task space-2D- Singularity
𝑑𝑟
𝜃
𝑑
𝑙1
𝑙2
x
y
x1
y1
−𝜃
o1
o 𝛼
𝛼 =𝜋
2−𝜃
2
𝑜1 =2𝑙
𝜃𝑐𝑜𝑠𝛼 2,
2𝑙
𝜃𝑐𝑜𝑠𝛼𝑠𝑖𝑛𝛼
𝑇
𝜃 =𝑙2 − 𝑙12𝑑
(𝑟 + 𝑑) =1
𝑘=
(𝑙1+𝑙2)𝑑
𝑙2−𝑙1
Recall: 𝑟 =1
𝑘=
2𝑙1𝑑
𝑙2−𝑙1
![Page 41: Kinematic Modelling of continuum robots following constant ...globex.coe.pku.edu.cn/file/upload/201807/05/0759057018.pdf · Definition of Constant Curvature • Shape of robot can](https://reader033.vdocuments.us/reader033/viewer/2022060320/5f0d18e17e708231d438aa0b/html5/thumbnails/41.jpg)
Compliant Robotics Peking University, Globex, July 201841
Mapping: Config to Task space-2D- better solution
𝑑𝑟
𝜃
𝑜1 = 𝑟 − 𝑟𝑐𝑜𝑠𝜃, 𝑟𝑠𝑖𝑛𝜃 𝑇
z
x
o1
𝑟 =1
𝑘=
(𝑙1+𝑙2)𝑑
𝑙2−𝑙1
![Page 42: Kinematic Modelling of continuum robots following constant ...globex.coe.pku.edu.cn/file/upload/201807/05/0759057018.pdf · Definition of Constant Curvature • Shape of robot can](https://reader033.vdocuments.us/reader033/viewer/2022060320/5f0d18e17e708231d438aa0b/html5/thumbnails/42.jpg)
Compliant Robotics Peking University, Globex, July 201842
In-plane translational vector p
Constant curvature in 3D
𝑑 ∙ cosΦ
𝑟
𝜃
𝑟𝑖
𝒑 = 𝑟 − 𝑟𝑐𝑜𝑠𝜃, 0, 𝑟𝑠𝑖𝑛𝜃 𝑇
z
x’
x’
⦿x
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Compliant Robotics Peking University, Globex, July 201843
Arc geometry
Shape of the robot can be expressed applying the transformation
Where Ri represent rotation matrices about axis i and
Mapping: Config to Task space-3D
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Compliant Robotics Peking University, Globex, July 201844
Mapping: Config to Task space-3D
𝑅𝑦 𝜃 =cos 𝜃 0 sin 𝜃0 1 0
−sin 𝜃 0 cos 𝜃𝑅𝑧 𝜙 =
cos𝜙 −sin𝜙 0sin𝜙 cos 𝜙 00 0 1
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Compliant Robotics Peking University, Globex, July 201845
Mapping: Config to Task space-3D
Aac= Aab Abc =Recap
𝑅𝑧 𝜙 =cos𝜙 −sin𝜙 0sin𝜙 cos𝜙 00 0 1
𝑅𝑦 𝜃 =cos 𝜃 0 sin 𝜃0 1 0
−sin 𝜃 0 cos 𝜃
Rt =cos𝜙𝑐𝑜𝑠𝜃 −sin 𝜃 𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜃sin𝜙𝑐𝑜𝑠𝜃 cos𝜙 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃
𝑝𝑡 =𝑐𝑜𝑠𝜙𝑟(1 − 𝑐𝑜𝑠𝜃)𝑠𝑖𝑛𝜙𝑟(1 − 𝑐𝑜𝑠𝜃)
𝑟𝑠𝑖𝑛𝜃
![Page 46: Kinematic Modelling of continuum robots following constant ...globex.coe.pku.edu.cn/file/upload/201807/05/0759057018.pdf · Definition of Constant Curvature • Shape of robot can](https://reader033.vdocuments.us/reader033/viewer/2022060320/5f0d18e17e708231d438aa0b/html5/thumbnails/46.jpg)
Compliant Robotics Peking University, Globex, July 201846
Mapping: Config to Task space-3D
Rt =cos𝜙𝑐𝑜𝑠𝜃 −sin 𝜃 𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜃sin𝜙𝑐𝑜𝑠𝜃 cos𝜙 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝜃 0 𝑐𝑜𝑠𝜃
𝑝𝑡 =𝑐𝑜𝑠𝜙𝑟(1 − 𝑐𝑜𝑠𝜃)𝑠𝑖𝑛𝜙𝑟(1 − 𝑐𝑜𝑠𝜃)
𝑟𝑠𝑖𝑛𝜃
T =
cos𝜙𝑐𝑜𝑠𝜃 −sin 𝜃 𝑐𝑜𝑠𝜙𝑠𝑖𝑛𝜃sin𝜙𝑐𝑜𝑠𝜃 cos𝜙 𝑠𝑖𝑛𝜙𝑠𝑖𝑛𝜃−𝑠𝑖𝑛𝜃0
00
𝑐𝑜𝑠𝜃0
𝑐𝑜𝑠𝜙𝑟(1 − 𝑐𝑜𝑠𝜃)𝑠𝑖𝑛𝜙𝑟(1 − 𝑐𝑜𝑠𝜃)
𝑟𝑠𝑖𝑛𝜃1
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Compliant Robotics Peking University, Globex, July 201847
Mapping: Config to Task space-3D
𝑘𝑙 = 𝜃
𝑑 ∙ cosΦ
𝑟
𝜃
𝑟𝑖
𝑟 =1
𝑘
S
θs
𝑘𝑠 = 𝜃𝑠
Let a point along the manipulator central
curve with arc length s
Arbitrary location on the continuum body
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Compliant Robotics Peking University, Globex, July 201848
The manipulator-independent mapping function from configuration to task space can be summarized by
Mapping: Config to Task space
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Compliant Robotics Peking University, Globex, July 201849
Recap: derivation from DH
• Same expression can be derived following DH convention
• Element along curve can be expressed by number of transformations
Mapping: Config to Task space
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Compliant Robotics Peking University, Globex, July 201850
• Inverse kinematic formulation is not trivial, particularly for multi-segment manipulators with large number of segments
• For Task to Configuration space mapping solution exist based on
– A closed geometric formulation (Neppalli et al. 2008)
– Jacobian derivation ( “differential kinematics”)
Inverse mapping
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Compliant Robotics Peking University, Globex, July 201851
• The Configuration to Actuator space mapping is similarly to FK highly individualized
• The solution exist based on
– A analytical geometric formulation
– Jacobian derivation ( “differential kinematics”)
Inverse mapping
![Page 52: Kinematic Modelling of continuum robots following constant ...globex.coe.pku.edu.cn/file/upload/201807/05/0759057018.pdf · Definition of Constant Curvature • Shape of robot can](https://reader033.vdocuments.us/reader033/viewer/2022060320/5f0d18e17e708231d438aa0b/html5/thumbnails/52.jpg)
Compliant Robotics Peking University, Globex, July 201852
Programming exercise
cc_1seg_execise_q.m
put cc_1seg_execise_q.m and transcc2D.m into the same folder
%tendon1
l1=12;
%tendon2
l2=8;
%pex is the coordinates of
unit vector of x axis in
the local frame of the tip
pex=[1 0]'
central bending axis of the arm
![Page 53: Kinematic Modelling of continuum robots following constant ...globex.coe.pku.edu.cn/file/upload/201807/05/0759057018.pdf · Definition of Constant Curvature • Shape of robot can](https://reader033.vdocuments.us/reader033/viewer/2022060320/5f0d18e17e708231d438aa0b/html5/thumbnails/53.jpg)
Compliant Robotics Peking University, Globex, July 201853
Try different combination of l1 and l2
What happens if l1=l2?
Testing the limit