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Dr. Ashish Dutta
Associate Professor
Dept. of Mechanical Engineering
Indian Institute of Technology Kanpur, INDIA
Humanoid robot design with softsole and springs for applicationsin optimal multi agent systems
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Evolution of multi agent systems fromwheeled robots to legged robots, UAVs, etc
Optimal multi agent system using mobilerobots
Design of energy optimal humanoid robot
with soft sole and springs as agents.
Other related research on exoskeleton,
orhtosis, etc.
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The idea of Multi agents (swarm robots) hasbeen borrowed from nature, emulating antcolonies, bee hives, etc.
The basic idea is to be able to perform acooperative task that cannot be performedby one individual robot.
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Fig. Mobile robot as agent.Fig. Two or more agents
pushing an object.
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Mobile robots coordination, net working,path planning.
Mobile robots were not necessarilyoptimized.
Mobile robots do not require balance and
are generally stable.
Newer types of agents: UAV, underwater
agents, bio-agents, humanoids, etc.
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Fig. Upper torso Humanoid that does not require balance.
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Multi Humanoid : System of system
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Fig. Quadrupeds agent Fig. Four legged and biped cooperation
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Wang and Kumar (2002) used the potentialfield method to obtain object closure using alarge number of robots.
Sugar and Kumar (1998) proposeddecentralized control of cooperating mobilemanipulators.
goal
object
mobilerobots
2. Optimal multi agent systems usingmobile robots
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A large number of robots have been used
(e.g. 100 or more) and capture points notoptimal.
Leader follower or potential field
approaches have been used for the robotmotion.
The dragging of the object is not optimized.
Obstacle avoidance is not considered.
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Why use a large number of robots if the task can bedone by few robots? Reduce energy consumption and Networking issues? Vary number of robots if required.
a. Optimal captureof a moving object using the
minimumnumber of robots.
b. Optimallypushing the captured object to the goal
point.
New contributions of our research
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Problem formulation: Given an object with n points on itsboundary, it is required to find the optimal grasp pointsfor satisfying form closure.
C.G.
1
6
13
20
Mobile Robot
10000100001000000100000010000Binary String
1 : Robot present
0 : Robot absent
GA based optimization using an objective function with
constraints.
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1cw ccwk
cw ccw
M Mf
N N N
+ = +
Prismatic Object
C.G.
1
6
13
20
Maximize
Moment is calculated byassuming unit normal forceapplied by robots.
N : Total No. of Robots
Mcw : Sum of CW moments
Mccw: Sum of CCW moments
k : parameter for
controlling No. of agents.
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Visibility angle () should be null
( ) ( ){ }
{ }1 2 .....
i
n
=
= i+1 i i i-1
r - r , r - r
Visibility angle is the common angle between allfreedom angles()
This constraint takes care of translation inaccessibility.
3
freedom
angles
1
31
No angle left uncover
4
2
4
2
3
freedom
angles
1
3
1
4
4
small angle left uncovered
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There should be robots creating moments in both the direction about C.G.
Feasible solution Non-feasible solution
C.G.
CW
CCW
CCW
C.G.
CW
CW
CW
This constraint takes care of rotation inaccessibility.
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- o wo ro o approac es e same e ge.
- Robots envelop never intersect with eachother on the object boundary.
Huge negative penalty (say -1e20) is imposed ifany of the constraint is violated.
Non-feasible solution
Object
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Simulation is carried out in MATLABTM using
the Genetic Algorithm.
A Binary String is used as design variable.
Population Size 52, Mutation probability0.12, Crossover Probability 0.8
Constraint violation attracts huge negative
penalty (say -1020).
Avg. simulation time is around 300 second,with Intel-P4 Machine.
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Fig. Results of simulation
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Parameter k inoptimization function isvaried to get differentsolution with different no.of robots for same object.
Variation of number of robots with constant K
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1 2 3.( ..... )
.
nf
f
= + + + +
=
D F F F F
D F
Desired
direction of
motion (D)
Robots participating in
pushing
Robots not sharing any load
Maximize
Fi
.. (1)
Problem formulation forpushing
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Gordy GA code is used to optimize eq (1)
724
19
29
16
13
1
000000000000000100100001000010
Binary String
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D
F3 F2F1
F4
D
F
F3 F2F1
F4
DF
F3 F2F1
F4
(a) (c)(b)
D
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- 1 0 - 5 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0
- 2 0
- 1 5
- 1 0
- 5
0
5
1 0
1 5
2 0
2 5
Copyright: Panka j Sharma , Dr.Anupam Sa xe na and Dr. Ashish Dutta, IITKanpur
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Non-holonomic mobile robots with twowheels independently driven weredeveloped.
Each mobile robot worked as an agentcontrolled by wireless communication withthe central computer.
Overhead vision based coordination.
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An algorithm was proposed for the optimalcapture and transfer of a moving object to adesired goal, using the minimum number ofmobile robots.
The advantages as compared to earlier methodsare that resources are minimized and it leads tolesser deadlocks and networking time.
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Biped: A two legged robot with 8 or moreDOF for walking .
GAIT: A fixed pattern of foot placements Trajectory: The path (optimal?)taken by the
free foot to come from the rear to the front .
3. Humanoids as agents
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Energy optimal trajectory generation forcooperation with human / robot / agent.
Energy savings considering deformation of softsole or ground on biped stability.
New design considering springs at the joints for
reduced energy consumption.
Modes of cooperation in Multi agent systems.
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Lateral plane Front plane Isometric view
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Using DAlembertsprinciple:
- Where mi is the
mass of each link and
ri are the positionvectors.
=++ 0)()( TGrXrrmipii
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The ZMP approach is used to find the stable configuration during theGAIT. The ZMP is the point where the sum of all the forces and the moment of all
the masses of the biped is zero. The ZMP moves forward in the direction of the locomotion.
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is the joint angle from xi-1axis to the xiaxis about thezi-1axis.
di is the distance from the origin of the (i-1)th coordinate frame to the intersection of the z i-1axis with thexi
axis along the zi-1axis.
ai is the shortestdistance between zi-1 and zi axes.
is the offset angle from the z i-1 axis to thezi axis about thexi axis.
i
i
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Euler-Lagrangian equation
where
L is Lagrangian function, and is given by
KE is total kinetic energy of the biped robot,
PE is potential energy of the biped robot,
is generalized coordinates of the robot arm,
is first time derivative of the generalized coordinates,
i is the torque applied to the system at joint into drive linki.
i = 1, 2, 3, . . . .8ii i
d L L fordt
=
i
i
8
1
i i
i
L KE PE
=
=
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( ) ( ) ( )..
( ) ,t = + +D h c
where , corresponds to inertial acceleration
related symmetric matrix
is the non-linear centrifugal force vector.
is the gravity loading force vector.
is the torque applied at joint
( )D
( ), h
( )c
( )t
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The dynamic equation can be further modified by taking the reaction force
into account
where
The component is given by
where
N2is calculated using
( ) ( ) ( )..
( ) , ( )t rt = + + D h c K t
1 2, 8( , ....... )Tt t t=t
(2) 2,3,....,7
(3) 1,8
i
i
i
for it
for i
== =
T
M
0
2( ) i=2,3,....7i i i for= T R a N
2( ) i=1 and 8i i for= M a N
Link i
80
1 ( )i ii m= + =2OA N r g 0
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Energy optimal trajectory is found using GA.
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Intermediate points are obtained using GA
i f
pq
tfTime (sec)
Joint Angle (rad)
3
0
1,2,.........8k j jk k
C t for j=
= =
where Cjk is constant.
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| | | |W dt =
Objective function
Work done involves
Finding the joint angle, the joint velocity and the joint acceleration. Finding Torques by using the dynamic equations
ZMP should always be inside the supporting polygon,
Free foot should never go inside the ground,
The hip joints should always move forward
The hip joints should not come below the specified height
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Variation of each angle is a cubic spline.
Given the starting and end points, two
intermediate points are decided.
Constraints -a) Foot should not go below the ground
b) ZMP should be inside the footc) Min height of the hip is specifiedd) Hip should move forward
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The total work done is minimized bychecking the constraints.
In case of violation of a constraint a large
penalty is added to the function value.
Using Lagrangian-Euler formulation,dynamic equations can be written as:
= dtddonework ||
( ) ( , ) ( ) D H C = + +
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8 DOF Robot Angle assignment
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Parameters:
Population Size :50
Crossover ratio:0.95 Mutation ratio:0.05
Iterations : 280
Link Length:0.25 m
Step Length:0.25 m
obstacle
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Parameters:
Population Size :50
Crossover ratio:0.95 Mutation ratio:0.05
Iterations : 190
Link Length:0.25 m
Step Length:0.25 m
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Trajectory are computed on assumptionthat sole and ground are perfectly rigid.
Effect of soft sole can be detrimental to
bipeds stability.
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If the force distributionis as shown:
Total Reaction Force
Centroid of trapezium =
( )1 22
d
F F= +
2 1
2 1
(2 )3( )
cgd F Fd
F F+=
+Frontal Plane
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By balancing the forces and momentsabout point B
Solving these equations, we get:
( )
( )( )
1 2
2 11 2
1 2
2(2 )
2 2 3
d F F F
d d d F F F T F F
F F
= +
+= + +
+
1 2
2 2
6
6
F TFd d
F TF
d d
= + =
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If the force distributionis as shown:
Reaction Force
Distance of centroid
1
2F d=
3
d
=
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Balancing forces and moments, we get:
Solving these equations
1
1
2
2 2 3
F dF
Fd F d d T
=
= +
1
32 2
32
FF
d TF
d Td
F
=
=
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Due to uneven forcedistribution, one sideof sole deforms
more, resulting in anangle
It effects as if one
more DOF
Intended becomes
+
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F = Kx ; where k= YA/ h
X = deformation, A= area of foot,h= thickness of sole, Y = modulus ofelasticity.
F1
F2
F
T
0c
0=g. orrec on
procedure for
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Iterative method
for finding new
Required Torque
(from dynamic model)corresponding sole deformation =
total angle due to deformation
error =
0d
= T
0 zmp zmp0
0 zmp 0d
+
zmp0c
0zmp 0d
0 zmp 0 zmp 0
d
- ( )+
= +
( )
error < 0.01
IF
zmpcorrected angle = 0
NO
YES
p
deformed
angle
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Material which givesfeasible new are
suitable for sole
For balance infrontal plane, new
for differentmaterials are:
YoungsModulus
N/mm2
Final valueof angle 1
(Starting
value =
25)
5000 -23.34
6000 26.0944
7000 25.3365
8000 25.1805
9000 25.0383
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EN/mm2
Actualangle
2(35.58
)
5000 -
34.8531
10000 -
35.21
77
50000 -
35.50
93
100000 -
35.54
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It has been proved that the deformation ofthe sole is affected mainly by the torque inthe first ankle joint.
This may be due to two reasons:
- The torque is maximum at this joint.
- The foot is longer in the lateral plane
for this robot.
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The trajectory has been computed in thefollowing steps:
a) Compute the ZMP and make correctiononly to the first joint.
b) Find the optimal trajectory for the otherjoints using GA, for a given step length.
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Fixed foot
Free foot
Energy per step withoutdeformation= 3.87 WEnergy per step withdeformation= 2.91 W
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8DOF Biped with Springs atthe joints
All the joints are revolute
The link length are equal except
for the hip link
The ankle has two DOFs
The knee has one DOF The hip joint has one DOF
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Design of a biped with torsionalsprings at the joints similar tobiological joints with compliance.
Determination of energy optimal
trajectories.
Determination of optimal stiffnessand reference angles of thesprings.
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( ) ( ) ( )..
( ) ,t = + +D h c
where ,
corresponds to inertial acceleration
related symmetric matrix
is the non-linear centrifugal force vector.
is the gravity loading force vector.
is the torque applied at joint
( )D
( ), h
( )c
( )t
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In this work the torsional spring acts as the energy absorber and absorb theenergy in the potential form.
The flexible joint can reduce the work done during the gait
Where
corresponds to the torsional stiffness of the spring
corresponds to the reference angle.
( ) ( ) ( )
..
( ) , ( )t r
t = + + D h c K
tK
r
1 2 8[ , ,....., ]Tt t t t K K K =K
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a) Optimal trajectory for the rigid robot withno spring
a) Optimal trajectory of robot with same
stiffness at each joint.
a) Optimal trajectory of robot with optimalindividual joint stiffness
a) Optimal trajectory for optimal instanceposition, individual stiffness at each joint,reference angle and initial orientation.
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Link length, L = 0.2 m Hip length, h = 0.14 m Step length = 0.2 m Mass matrix = [0 1.3122 1.3122 2.7128 1.3122 1.3122 0 1.3348] Kg Length of the foot = 0.2 m Width of the foot = 0.07 m Without loss of generality the fixed foot is taken as the origin.
The hip height has taken to be 0.32m. The initial tilt of the biped is taken to be pi/9 and fixed all along the
simulation The distance of the free foot and hip joint is 0.1m and 0.05m on both the
side Reference position of the torsion spring is [0 0 /2 /2 3 /2 - /
2 0 0]rad
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K=0.25 Nm/rad, WD=5.4140 Watt K=0.5 Nm/rad, WD=4.3976 Watt K=1 Nm/rad, WD=5.7027 Watt
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K=1.5 Nm/rad, 6.1584 Watt K=2 Nm/rad, WD=8.4929 Watt K=3 Nm/rad, WD=10.7429 Watt
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K=0.25 Nm/rad, WD=5.6363 WattK=0.5 Nm/rad, WD=3.9654 Watt K=1 Nm/rad, WD=4.1297 Watt
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K=1.5 Nm/rad, WD=4.2784 Watt K=2 Nm/rad, WD=6.6247 WattK=3 Nm/rad, WD=7.4129 Watt
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K=0.25 Nm/rad, WD=4.1442 Watt K=0.5 Nm/rad, WD=3.5275 Watt K=1 Nm/rad, WD=4.6562 Watt
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K=1.5 Nm/rad, WD=4.6793 Watt K=2 Nm/rad, WD=4.9295 Watt K=3 Nm/rad, WD=8.4450 Watt
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The red, blue and green colored graph shows the variation of the work done with the
time step of 0.5, 1 and 2 sec respectively.
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GOALTo determine the optimal individual joint stiffness for obtainingenergy optimal gait
GA Parameters
Population size was set to 200, Maximum number of iterations was set to 4000,
Crossover probability = 0.95, and
Mutation probability = 0.05.
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WD=3.8577 WattWD=3.4036 Watt WD=5.8366 Watt
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Table 6.1:Stiffness at different joints
Joint No. 1 2 3 4 5 6 7 8
Stiffness
(Nm/rad)
Set 1 0.3838 1.3471 2.7807 2.1294 0.2171 0.7128 0.5361 2.8608
Set 2 2.4627 0.2471 2.1926 0.9228 0.5223 0.6539 2.4460 1.7384
Set 3 2.8773 0.5322 1.9642 1.5793 0.5955 0.2332 2.7530 0.7708
Average 1.9079 0.7088 2.3125 1.5438 0.4450 0.5333 1.9117 1.7900
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GOAL: To Determine Individual joint stiffness
The reference position of the torsional spring
The initial and final stances
The tilt angle of the biped robot GA parameters
Set no. Population size Maximum number of iterations
Crossoverprobability
Mutationprobability
Set1 50 4000 0.95 0.05
Set2 500 2000 0.95 0.05
Set3 200 2000 0.95 0.05
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WD=2.1605 Watt WD=1.8273 Watt WD=2.28Watt
Optimal stiffness of torsional springs at different joints
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Stiffness at different joints
Joint No. 1 2 3 4 5 6 7 8
Stiffness(Nm/rad)
Set 1 0.9587 1.7148 0.8393 2.9919 2.7364 0.6546 1.1190 2.1605
Set 2 2.65 1.4582 0.1581 2.5154 1.5139 1.0267 1.0901 1.6588
Set 3 2.3227 0.2807 0.7013 0.0953 2.8591 2.8329 1.2758 0.2723
Average 1.9771 1.1512 0.5662 1.8675 2.3698 1.5047 1.1616 1.3638
Optimal reference angles at different joints
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Reference angles at different joints
Joint No. 1 2 3 4 5 6 7 8
Angle
(rad)
Set 1 -0.0561 -1.2783 1.8674 -0.98 3.5185 -0.3786 1.1394 -0.3239
Set 2 -0.0082 -1.0817 1.8844 -0.8675 3.6571 -0.4165 1.5070 -0.1918
Set 3 -0.1075 -1.5424 1.9564 -0.0056 3.3750 -0.9179 0.9242 -0.3080
Average -0.057 -1.301 1.902 -0.206 3.5169 -0.571 1.1902 -0.275
Optimal Stance for T=1 sec
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Free Foot Hip
Initial Stance (m)
a
Final Stance (m)
b
Initial Stance (m)
c
Final Stance (m)
d
Height
(m)
Set 1 -0.0948 0.1072 -0.0561 0.0502 0.2978
Set 2 -0.0805 0.0852 -0.0469 0.0402 0.3060
Set 3 -0.0966 0.1090 -0.0512 0.0467 0.2928
Average -0.0906 0.1004 -0.514 0.0457 0.298
Initial Stance and Final Stance Position
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Initial Orientation (1) in radian
Set 1 -0.4397
Set 2 -0.4201
Set 3 -0.4510
Average -0.437
Initial Orientation of Biped
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As the time for GAIT increases from 0.5 sec to 2 sec thework done reduces from 6.9538 Watt to 4.3495 Watt for
rigid case.
For a time step of 0.5 sec, 1 sec and 2 sec the minimum
work done is 4.3976 Watt, 3.9654 Watt and 3.5275 Watt
respectively when the stiffness at each joint is 0.5 Nm/rad
For optimal stiffness at each joint the minimum work done
is 3.4036 Watt.
Optimizing all the parameters namely stance position,
stiffness at each joint, reference angle and initial tilt the
minimum work done is 1.8273 Watt
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1. Performing a task in coordination (withforce /moment interaction)
2. Performing a task in formation (no forces /
moments involved)
3. Others (interacting with people using vision,
speech etc.).
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Concept of ZMP is valid only for one biped.
In case of multiple bipeds the conditions ofbalance are not very clear.
One idea is to ensure that the ZMP is inbetween all the biped and environment
contact points.
What happens if one robot looses contact?
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The system is balanced as long as the ZMP isinside the common feet polygons.
Sensory information determines position of
the common object / environment (inclinationor vision sensors)
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Major concern is the time of convergence :days !
Interaction is still highly constrained.
Future is to develop real time controlalgorithms.
Finger exoskeleton for rehabilitationf k i
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of stroke patients
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Design optimal mechanical system Use EMG from muscles Use EEG from brain as a switch foractivation.
Proximal Phalanx Four Bar
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DEPARTMENT OF ELECTRICAL ENGINEERING
Middle Phalanx Four Bar
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DEPARTMENT OF ELECTRICAL ENGINEERING
Distal Phalanx Four Bar
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DEPARTMENT OF ELECTRICAL ENGINEERING
Index Finger Four Bar
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DEPARTMENT OF ELECTRICAL ENGINEERING
Index Finger Four Bar
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DEPARTMENT OF ELECTRICAL ENGINEERING
Objective:
To design the Optimized Exoskeleton for the Index finger based on Four BarDesign using Path Generation
GA Optimization Details:
Iterations: 4000 for each four bar
Population Size: 100
Generation: 700
Cross-Over Fraction: 0.8
Proximal Phalanx Four Bar (contd)
Error = 1.9 cm2
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DEPARTMENT OF ELECTRICAL ENGINEERING
Middle Phalanx Four Bar (contd)
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DEPARTMENT OF ELECTRICAL ENGINEERING
Distal Phalanx Four Bar (contd)
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DEPARTMENT OF ELECTRICAL ENGINEERING
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Reconfigurable prosthetic socket
)heel contact (b)mid stance (c) toe off
extensionmoment at the
groundreaction force
A
CB
D
Fig.1 Diagram of walking cycle with lowerlimb prosthesis& Design concept of the MR
Fig.14
Two main problems:
1. Misalignment
2. Change in stump
size
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Fig. 15 Correction of misalignment of the socket.
Instrumented socket with
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Instrumented socket with
slip and force sensors
Slip
Sensors
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VAS 0 1 1 2
2 3
3 4 4 5 5 6
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(b) 63[kPa]
(a) 38[kPa]
PT(L) 0.72
PT(C) 0.63
PT(M) 1.54HF 1.81
TL 1.72
TC 1.09
TM 1.54
TL 1.54
TC 1.45TM 1.63
TL 1.54
TC 1.81
TM 1.36
FP 3.81
FP 3.36FP 3.36
FP 3.18
Average 1.89
PT(L) 0.54
PT(C) 0.36
PT(M) 0.54
HF 0.72
TL 0.72
TC 0.54
TM 0.81
TL 0.54
TC 0.72
TM 0.90
TL 0.72
TC 1.09
TM 1.00
FP 2.27
FP 2.18
FP 2.45
FP 1.90
Average 1.37
6 7 7 8
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(a) (b)
patellar tendon
tibial crest
fossa poplitea
stump end
tube fittings
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0
5
10
15
0 50 100 150
pressure [kPa]
displacemen
t[mm]
0[T]
0.12[T]
0.12[T]2
0.12[T]3
0.12[T]40[T]
0.12[T]
0.24[T]
0.36[T]
0.48[T]Rigid
socket
Soft
Flexible
socket
Active
socket
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Two rovers and a Lander on the moon.
Navigation
Kinematics , Dynamics and Control.
System of systems
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