Prof. Alessandro De Luca - uniroma1.itdeluca/rob2_en/14...Robotics 2 2 sensors: position (encoders)...
Transcript of Prof. Alessandro De Luca - uniroma1.itdeluca/rob2_en/14...Robotics 2 2 sensors: position (encoders)...
Robot-environment interactiona robot (end-effector) may interact with the environment
ยง modifying the state of the environment (e.g., pick-and-place operations)ยง exchanging forces (e.g., assembly or surface finishing tasks)
control of compliant motion
sensors: as before +6D force/torque (at the robot wrist)
โpeg in holeโtask
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sensors: position (encoders) at the joints* or vision at the Cartesian level
S
G
control of free motion
inspectiontask
*and velocity (by numerical differentiation or, more rarely, with tachos)
Robot compliance
PASSIVE robot end-effector equipped with mechatronic devices that โcomplyโ with the generalized forces applied at the TCP = Tool Center Point
ACTIVErobot is moved by a control law so as to react in a desired way to generalized forces applied at the TCP (typically measured by a F/T sensor)
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RCC = Remote Center of Compliance device
ยง admittance controlcontact forces โ velocity commands
ยง stiffness/compliance controlcontact displacements โ force commands
ยง impedance controlcontact displacements โ contact forces
Typical evolution of assembly forces
chamfer angle ๐ฝ = to ease the insertion, related also to the tolerances of the hole
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โpeg-in-holeโ task
Tasks with environment interactionn mechanical machining
n deburring, surface finishing, polishing, assembly,...n tele-manipulation
n force feedback improves performance of human operators in master-slave systems
n contact exploration for shape identificationn force and velocity/vision sensor fusion allow 2D/3D geometric
identification of unknown objects and their contour followingn dexterous robot hands
n power grasp and fine in-hand manipulation require force/motion cooperation and coordinated control of the multiple fingers
n cooperation of multi-manipulator systemsn the environment includes one of more other robots with their own
dynamic behaviorsn physical human-robot interaction
n humans as active, dynamic environments that need to be handled under full safety premises โฆ
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Abrasive finishing of surfaces
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video
technological processes: cold forging of surfacesand hammer peening by pneumatic machine
Non-contact surface finishing
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Fluid Jet technology
Pulsed Laser technology
video
video
H2020 EU project for theFactory of the Future (FoF)
In all cases ...
n for physical interaction tasks, the desired motion specification and execution should be integrated with complementary data for the desired force
รจ hybrid planning and control objectivesn the exchanged forces/torques at the contact(s) with the
environment can be explicitly set under control or simply kept limited in an indirect way
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Evolution of control approachesa bit of history from the late 70โs-mid โ80s โฆ
n explicit control of forces/torques only [Whitney]n used in quasi-static operations (assembly) in order to avoid deadlocks
during part insertionn active admittance and compliance control [Paul, Shimano, Salisbury]
n contact forces handled through position (stiffness) or velocity (damping) control of the robot end-effector
n robot reacts as a compressed spring (with damper) in selected/all directionsn impedance control [Hogan]
n a desired dynamic behavior is imposed to the robot-environment interaction, e.g., a โmodelโ with forces acting on a mass-spring-damper
n mimics the human arm behavior moving in an unknown environmentn hybrid force-motion control [Mason]
n decomposes the task space in complementary sets of directions where either force or motion is controlled, based onn a purely kinematic robot model [Raibert, Craig]n the actual dynamic model of the robot [Khatib]
appropriate for fast and accurate motion in dynamic interaction...Robotics 2 16
Interaction tasks of interest
interaction tasks with the environment that requiren accurate following/reproduction by the robot end-effector of desired
trajectories (even at high speed) defined on the surface of objectsn control of forces/torques applied at the contact with environments
having low (soft) or high (rigid) stiffness
robot
turninga crank
e.g., opening a door
deburring task
e.g., removing extra glue fromthe border of a car windshield
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Impedance vs. Hybrid control
n environment = mechanical system undergoing small but finite deformations
n contact forces arise as the result of a balance of two coupled dynamic systems (robot+environment)
รจ desired dynamic characteristics are assigned to the force/motion interaction
n a rigid environment reduces the degrees of freedom of the robot when in (bi-/uni-lateral) contact
n contact forces result from attempts to violate geometric constraints imposed by the environment
รจ task space is decomposed in sets of directions where only motion or only reaction forces are feasible
environment model ( domain of control application)impedance control hybrid force/motion control
ยง the required level of knowledge about the environment geometry is only apparently different between the two control approaches
ยง however, measuring contact forces may not be needed in impedance control, while it always necessary in hybrid force/motion control
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Impedance vs. Hybrid controln opening a door with a mobile
manipulator under impedance control
n piston insertion in a motor based on hybrid control of force-position (visual)
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video video
รน
A typical constrained situation โฆ
robot
wrist6D F/T sensor
or RCC (or both)tool workpiece
(rigid)
the robot end-effector follows in a stable and accurateway the geometric profile of a very stiff workpiece,
while applying a desired contact force
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An unusual compliant situation โฆ
Trevelyan (AUS): Oracle robotic system in a test dated 1981โฆis the sheep happy?
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A mixed interaction situation
processing/reasoning on force measurements leads to a sequence of fine motions
โ correct completion of insertion task withhelp of (sufficiently large) passive compliance
1. approach 2. search 3. insertion
X,Y-axescontrol
Z-axiscontrol
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Ideally constrained contact situation
๐๐&
๐'
๐(
๐ฅ( = ๐
๐( = โ๐&
โidealโ = robot (sketched here as a Cartesian mass)+ environment are both infinitely STIFF(and without friction at the contact)
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a first possible modeling choice for very stiff environments
๐ฅ < ๐ ๐( = 0
๐ฅ = ๐๐& โฅ 0
๏ฟฝฬ๏ฟฝ = 0๏ฟฝฬ๏ฟฝ = 0
๐๏ฟฝฬ๏ฟฝ = ๐& + ๐(๐๏ฟฝฬ๏ฟฝ = ๐'
In more complex situations
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n how can we describe more complex contact situations, where the end-effector of an articulated robot (not yet reduced to a Cartesian mass via feedback linearization control) is constrainedto move on an environment surface with nonlinear geometry?
n example: a planar 2R robot with end-effector moving on a circle
end-effectorconstrained on
a circular surface
(๐ฅ5, ๐ฆ5)๐
๐ฅ
๐ฆ
Constrained robot dynamics - 1
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n suppose that the task variables are subject to ๐ < ๐ (bilateral) geometric constraints in the general form ๐ ๐ = 0 and define
โ ๐ = ๐ ๐ ๐ = 0
n the constrained robot dynamics can be derived using again the Lagrange formalism, by defining an augmented Lagrangian as
where the Lagrange multipliers ๐ (a ๐-dimensional vector) can be interpreted as the generalized forces that arise at the contact when attempting to violate the constraints
๐ฟA ๐, ๏ฟฝฬ๏ฟฝ, ๐ = ๐ฟ ๐, ๏ฟฝฬ๏ฟฝ + ๐Bโ ๐ = ๐ ๐, ๏ฟฝฬ๏ฟฝ โ ๐ ๐ + ๐Bโ ๐
n consider a robot in free space described by its Lagrange dynamic model and a task output function (e.g., the end-effector pose)
๐ ๐ ๏ฟฝฬ๏ฟฝ + ๐ ๐, ๏ฟฝฬ๏ฟฝ + ๐ ๐ = ๐ข ๐ = ๐(๐) ๐ โ โI
Constrained robot dynamics - 2
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contact forces doNOT produce work
n applying the Euler-Lagrange equations in the extended space of generalized coordinates ๐ AND multipliers ๐ yields
๐๐๐ก
๐๐ฟA๐๏ฟฝฬ๏ฟฝ
Bโ
๐๐ฟA๐๐
B=๐๐๐ก
๐๐ฟ๐๏ฟฝฬ๏ฟฝ
Bโ
๐๐ฟ๐๐
Bโ
๐๐๐ ๐Bโ(๐)
B
= ๐ข
๐๐ฟA๐๐
B= โ ๐ = 0
๐ด ๐ =๐โ(๐)๐๐
where we defined the Jacobian of the constraints as the matrixsubject to
that will be assumed of full row rank (= ๐)
(โ )๐ ๐ ๏ฟฝฬ๏ฟฝ + ๐ ๐, ๏ฟฝฬ๏ฟฝ + ๐ ๐ = ๐ข + ๐ดB(๐)๐
โ ๐ = 0
Constrained robot dynamics - 3
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n we can eliminate the appearance of the multipliers as followsn differentiate the constraints twice w.r.t. time
n substitute the joint accelerations from the dynamic model (โ )(dropping dependencies)
n solve for the multipliers
to be replaced in the dynamic model...
the inertia-weighted pseudoinverse of the constraint Jacobian ๐ด
invertible ๐ร๐ matrix,when ๐ด is full rank constraint
forces ๐ areuniquely
determinedby the robotstate (๐, ๏ฟฝฬ๏ฟฝ)
and input ๐ข !!
โ ๐ = 0 โ โฬ =๐โ(๐)๐๐ ๏ฟฝฬ๏ฟฝ = ๐ด ๐ ๏ฟฝฬ๏ฟฝ = 0 โ โฬ = ๐ด ๐ ๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ ๐ ๏ฟฝฬ๏ฟฝ = 0
๐ด๐OP ๐ข + ๐ดB๐ โ ๐ โ ๐ + ๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ = 0
๐ = (๐ด๐OP๐ดB)OP ๐ด๐OP ๐ + ๐ โ ๐ข โ ๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ= ๐ดQ#
B ๐ + ๐ โ ๐ข โ ๐ด๐OP๐ดB OP๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ
Constrained robot dynamics - 4
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n the final constrained dynamic model can be rewritten as
where ๐ดQ# ๐ = ๐OP(๐)๐ดB(๐)(๐ด(๐)๐OP(๐)๐ดB(๐))OP and with
n if the robot state (๐(0), ๏ฟฝฬ๏ฟฝ(0)) at time ๐ก = 0 satisfies the constraints, i.e.,
then the robot evolution described by the above dynamics will be consistent with the constraints for all ๐ก โฅ 0 and for any ๐ข(๐ก)
ยง this is a useful simulation model (constrained direct dynamics)
dynamically consistent projection matrix
โ ๐ 0 = 0, ๐ด ๐ 0 ๏ฟฝฬ๏ฟฝ(0) = 0
๐ = ๐ดQ# (๐)B ๐(๐, ๏ฟฝฬ๏ฟฝ) + ๐(๐) โ ๐ข โ ๐ด ๐ ๐OP ๐ ๐ดB ๐ OP๏ฟฝฬ๏ฟฝ(๐)๏ฟฝฬ๏ฟฝ
๐ ๐ ๏ฟฝฬ๏ฟฝ = ๐ผ โ ๐ดB(๐) ๐ดQ# (๐)B ๐ข โ ๐(๐, ๏ฟฝฬ๏ฟฝ โ ๐(๐)) โ๐(๐)๐ดQ# (๐)๏ฟฝฬ๏ฟฝ(๐)๏ฟฝฬ๏ฟฝ
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Example โ ideal massconstrained robot dynamics
๐๐&
๐'
๐(
๐ฅ = ๐
๐ =๐ฅ๐ฆ
๐ข =๐&๐' ๐ = ๐ 0
0 ๐
๐๏ฟฝฬ๏ฟฝ = ๐ข robot dynamics in free motion
๐ผ โ ๐ดB(๐) ๐ดQ# (๐)B = 0 0
0 1dynamically consistent
projection matrix
constrainedrobot dynamics๐ ๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ = ๐๏ฟฝฬ๏ฟฝ = 0 00 1 ๐ข =
0๐'
๐ = โ ๐ดQ# (๐)B๐ข = โ 1 0 ๐ข = โ๐& multiplier (contact force ๐()
โ ๐ = ๐ฅ โ ๐ = 0 ๐ด ๐ = 1 0 ๐ดQ# ๐ = โฏ = 10โ โ
Example โ planar 2R robotconstrained robot dynamics
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๐1
๐2
๐1๐2
(๐ฅ5, ๐ฆ5)๐
๐ฅ
๐ฆ๐ ๐ = ๐ฅ โ ๐ฅ5 X + ๐ฆ โ ๐ฆ5 X โ ๐ X = 0
๐ =๐ฅ๐ฆ
๐ = ๐ ๐ = ๐P cos ๐P + ๐X cos ๐P + ๐X๐P sin ๐P + ๐X sin ๐P + ๐X
โ ๐ = ๐(๐ ๐) =๐P cos ๐P + ๐X cos ๐P + ๐X โ ๐ฅ5 X + ๐P sin ๐P + ๐X sin ๐P + ๐X โ ๐ฆ5 X โ ๐ X = 0
โฬ =๐๐๐๐
๐๐๐๐ ๏ฟฝฬ๏ฟฝ =
2 ๐ฅ โ ๐ฅ5 2 ๐ฆ โ ๐ฆ5 ๐ฝ_(๐)๏ฟฝฬ๏ฟฝ
= 2 ๐P๐P + ๐X๐PX โ ๐ฅ5 2 ๐P๐ P + ๐X๐ PX โ ๐ฆ5 ๐ฝ_ ๐ ๏ฟฝฬ๏ฟฝ = ๐ด(๐)๏ฟฝฬ๏ฟฝ
Reduced robot dynamics - 1
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n by imposing ๐ constraints โ(๐) = 0 on the ๐ generalized coordinates ๐, it is also possible to reduce the description of the constrained robot dynamics to a ๐ โ๐ dimensional configuration space
n start from constraint matrix ๐ด(๐) and select a matrix ๐ท(๐) such that
n define the (๐ โ๐)-dimensional vector of pseudo-velocities ๐ฃ as the linear combination (at a given ๐) of the robot generalized velocities
n inverse relationships (from โpseudoโ to โgeneralizedโ velocities and accelerations) are given by
properties of block products in inverse matrices have been used for eliminating the appearance of ๏ฟฝฬ๏ฟฝ (often ๐น is only known numerically)
is a nonsingular๐ร๐ matrix
๐ด(๐)๐ท(๐)
๐ด(๐)๐ท(๐)
OP= ๐ธ(๐) ๐น(๐)
๐ฃ = ๐ท(๐)๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ = ๐ท ๐ ๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ(๐)๏ฟฝฬ๏ฟฝ
๏ฟฝฬ๏ฟฝ = ๐น ๐ ๐ฃ ๏ฟฝฬ๏ฟฝ = ๐น ๐ ๏ฟฝฬ๏ฟฝ โ ๐ธ ๐ ๏ฟฝฬ๏ฟฝ ๐ + ๐น(๐)๏ฟฝฬ๏ฟฝ(๐) ๐น ๐ ๐ฃ
Reduced robot dynamics โ 2whiteboard โฆ
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๐ด(๐)๐ท(๐)
OP= ๐ธ(๐) ๐น(๐) a number of properties from this definitionโฆ
two matrix inverse products๐ด(๐)๐ท(๐) ๐ธ(๐) ๐น(๐) = ๐ด ๐ ๐ธ(๐) ๐ด ๐ ๐น(๐)
๐ท ๐ ๐ธ(๐) ๐ท ๐ ๐น(๐) =๐ผQรQ 00 ๐ผ(IOQ)ร(IOQ)
๐ธ(๐) ๐น(๐) ๐ด(๐)๐ท(๐) = ๐ธ ๐ ๐ด ๐ + ๐น ๐ ๐ท ๐ = ๐ผIรI
from pseudo-velocity ๐ฃ = ๐ท(๐)๏ฟฝฬ๏ฟฝsince ๐น is a right inverse of thefull row rank matrix ๐ท (๐ท๐น = ๐ผ)
๏ฟฝฬ๏ฟฝ = ๐น ๐ ๐ฃ= ๐ทB(๐) ๐ท(๐)๐ทB(๐) OP๐ฃ
(in fact๐ท๏ฟฝฬ๏ฟฝ = ๐ท๐น๐ฃ
= ๐ฃ)
differentiating w.r.t. time ๏ฟฝฬ๏ฟฝ = ๐น ๐ ๐ฃ
๏ฟฝฬ๏ฟฝ = ๐น๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ๐ฃ = ๐น๏ฟฝฬ๏ฟฝ + (๏ฟฝฬ๏ฟฝ๐ท)๏ฟฝฬ๏ฟฝ
three useful identities!
differentiating w.r.t. time ๏ฟฝฬ๏ฟฝ๐ด + ๐ธ๏ฟฝฬ๏ฟฝ + ๏ฟฝฬ๏ฟฝ๐ท + ๐น๏ฟฝฬ๏ฟฝ = 0 โ
= ๐น(๐)๏ฟฝฬ๏ฟฝ โ ๐ธ ๐ ๏ฟฝฬ๏ฟฝ ๐ + ๐น ๐ ๏ฟฝฬ๏ฟฝ ๐ ๐น(๐)๐ฃ
(โ)= ๐น๏ฟฝฬ๏ฟฝ โ ๏ฟฝฬ๏ฟฝ๐ด + ๐ธ๏ฟฝฬ๏ฟฝ + ๐น๏ฟฝฬ๏ฟฝ ๐น๐ฃ
0๐ผ
๏ฟฝฬ๏ฟฝ๐ท
Reduced robot dynamics - 3
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n consider again the dynamic model (โ ), dropping dependencies
n since ๐ด๐ธ = ๐ผ, multiplying on the left by ๐ธB isolates the multipliers
n since ๐ด๐น = 0, multiplying on the left by ๐นB eliminates the multipliers
n substituting in the latter the generalized accelerations and velocities with the pseudo-accelerations and pseudo-velocities leads finally to
which is the reduced (๐ โ๐)-dimensional dynamic modeln similarly, the expression of the multipliers becomes
invertible ๐ โ๐ ร ๐ โ๐
positive definite matrix
๐๏ฟฝฬ๏ฟฝ + ๐ + ๐ = ๐ข + ๐ดB๐
๐ธB ๐๏ฟฝฬ๏ฟฝ + ๐ + ๐ โ ๐ข = ๐
๐นB๐๏ฟฝฬ๏ฟฝ = ๐นB ๐ข โ ๐ โ ๐
๐นB๐๐น ๏ฟฝฬ๏ฟฝ = ๐นB ๐ข โ ๐ โ ๐ +๐ ๐ธ๏ฟฝฬ๏ฟฝ + ๐น๏ฟฝฬ๏ฟฝ ๐น๐ฃ
(ยง)๐ = ๐ธB ๐๐น๏ฟฝฬ๏ฟฝ โ๐ ๐ธ๏ฟฝฬ๏ฟฝ + ๐น๏ฟฝฬ๏ฟฝ ๐น๐ฃ + ๐ + ๐ โ ๐ข
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Example โ ideal massreduced robot dynamics
๐๐&
๐'
๐(
๐ฅ = ๐
๐ =๐ฅ๐ฆ
๐ข =๐&๐'
๐ = ๐ 00 ๐
๐๏ฟฝฬ๏ฟฝ = ๐ข robot dynamics in free motion
โ ๐ = ๐ฅ โ ๐ = 0 ๐ด = 1 0โ โ ๐ด๐ท = 1 0
0 1 = ๐ธ ๐น
reducedrobot dynamics๐นB๐๐น ๏ฟฝฬ๏ฟฝ = 0 1 ๐ 0
0 ๐01 ๏ฟฝฬ๏ฟฝ = ๐๏ฟฝฬ๏ฟฝ = ๐' = ๐นB๐ข
multiplier(contact force ๐()
๐ = ๐ธB ๐๐น๏ฟฝฬ๏ฟฝ โ ๐ข
= 1 0 ๐ 00 ๐
01 ๏ฟฝฬ๏ฟฝ โ
๐&๐'
= โ 1 0๐&๐'
= โ๐&
๐ฃ = ๐ท๏ฟฝฬ๏ฟฝ = ๏ฟฝฬ๏ฟฝ pseudo-velocity
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a feasible selection of matrix ๐ท(๐)
robotJacobian
out of robotsingularities
๐1
๐2
๐1๐2
(๐ฅ5, ๐ฆ5)๐
๐ฅ
๐ฆ๐ ๐ = ๐ฅ โ ๐ฅ5 X + ๐ฆ โ ๐ฆ5 X โ ๐ X = 0
๐ =๐ฅ๐ฆ
๐ฃ = (scalar) value of end-effector velocity reduced
along the tangentto the constraint
๐ด ๐ = 2 ๐ฅ โ ๐ฅ5 2 ๐ฆ โ ๐ฆ5 ๐ฝ_ ๐= 2 ๐P๐P + ๐X๐PX โ ๐ฅ5 2 ๐P๐ P + ๐X๐ PX โ ๐ฆ5 ๐ฝ_ ๐
๐ท(๐) = โ12๐ฆ โ ๐ฆ5
12๐ฅ โ ๐ฅ5 ๐ฝ_ ๐ det
๐ด(๐)๐ท(๐) = ๐ X h det ๐ฝ_(๐) โ 0
๐ด(๐)๐ท(๐)
OP= ๐ธ(๐) ๐น(๐)
a scalar๐ฃ = ๐ท(๐)๏ฟฝฬ๏ฟฝ ๏ฟฝฬ๏ฟฝ = ๐น ๐ ๐ฃ = ๐ฝ_OP(๐)
jโ2(๐ฆ โ ๐ฆ5)๐ X
j2(๐ฅ โ ๐ฅ5)๐ X
๐ฃ
Example โ planar 2R robotreduced robot dynamics
Control based on reduced robot dynamics
Robotics 2 37
n the reduced ๐ โ ๐ dynamic expressions are more compact but also more complex and less used for simulation purposes than the ๐-dimensional constrained dynamics
n however, they are useful for control design (reduced inverse dynamics)n in fact, it is straightforward to verify that the feedback linearizing
control law
applied to the reduced robot dynamics and to the expression (ยง) of the multipliers leads to the closed-loop system
๐ข = ๐ + ๐ โ๐ ๐ธ๏ฟฝฬ๏ฟฝ + ๐น๏ฟฝฬ๏ฟฝ ๐น๐ฃ +๐๐น๐ขP โ ๐ดB๐ขX
๏ฟฝฬ๏ฟฝ = ๐ขP ๐ = ๐ขX
Note: these are exactly in the form of the ideal mass example of slide #24, with ๐ฃ = ๏ฟฝฬ๏ฟฝ, ๐ขP = ๐'/๐, ฮป = ๐(, ๐ขX = โ๐& (being ๐ = 2, ๐ = 1, ๐ โ ๐ = 1)
Compliant contact situation
๐พ(
Robotics 2 38
a second possible modeling choice for softer environments
๐ฅ < ๐ ๐( = 0๐ฅ โฅ ๐ ๐( = ๐พ((๐ฅ โ ๐)
๐๏ฟฝฬ๏ฟฝ = ๐& + ๐(๐๏ฟฝฬ๏ฟฝ = ๐'
๐๐&
๐'
๐(
๐ฅ( = ๐
compliance/impedancecontrol (in all directions) is here a good choicethat allows to handleยง uncertain positionยง uncertain orientationof the wall
with ๐พ( > 0 being the stiffness of the environment
Robot-environment contact types modeled by a single elastic constant
๐พ = ๐พo
rigid environmentcompliantforce sensor
compliant environment
rigid robot(including
force sensor) ๐พ = ๐พ(
๐พo
๐พ(
negligible intermediatemass
Robotics 2 39
series of springs =sum of compliances
(inverse of stiffnesses)
1๐พ =
1๐พo+1๐พ(
๐พ =๐พo๐พ(๐พo +๐พ(
Force control 1-dof robot-environment linear dynamic models
Robotics 2 40
n with a force sensor (having stiffness ๐๐ and damping ๐๐ ) measuring the contact force ๐น๐n stability analysis of a proportional control loop for regulation of the contact force (to a
desired constant value ๐น๐) can be made using the root-locus method (for a varying ๐๐)n by including/excluding work-piece compliance and/or robot (transmission) compliance
(see the paper by S. Eppinger, W.P. Seering: IEEE Control Systems Mag., 1987;available as extra material on the course web)
+ work-piececompliance
+ robot compliance
large ๐๐โ
unstable
stablefor all ๐๐
๐นr = ๐o๐ฅ_ ๐นr = ๐o(๐ฅ_โ๐ฅs)
๐นr = ๐X(๐ฅXโ๐ฅs)๐นr = ๐o๐ฅX
๐น = ๐t ๐นu โ ๐นr , ๐t> 0
Tasks requiring hybrid control
the robot should turn a crankhaving a free-spinning handle
Robotics 2 41
two generalized directions of instantaneousfree motion
at the contact:tangential velocity& angular velocityaround handle axis
โfour directions of generalized reaction forcesat the contact
Tasks requiring hybrid control
Robotics 2 42
the robot should turn a crankhaving a fixed handle
one direction onlyof instantaneous
free motionat the contact:
tangential velocity
โfive directions of generalized reaction forcesat the contact
Tasks requiring hybrid control
the robot should push a mass elastically coupled to a wall and constrained in a guide
Robotics 2 43
Tasks requiring hybrid control
generalized hybrid modeling and control for dynamic environmentsA. De Luca, C. Manes: IEEE Trans. Robotics and Automation, vol. 10, no. 4, 1994
Robotics 2 44
direction of free motion control(no contact forces can be imposed)
direction of contact force control (no motion can be imposed)
dynamic direction of control:either motion is controlled
(and a contact force results)or contact force is controlled
(and a motion results)
KE
three sets of possible directions in the task frame