Statics Statics Basilio Bona 1 ROBOTICA 03CFIOR
Statics – 1 � Statics studies the relations between the task space forces/torques and the joint forces/torques in static equilibrium conditions � The former derive from possible interactions with the environment (e.g., when the TCP pushes against a surface) environment (e.g., when the TCP pushes against a surface) � The latter are due to the power provided by joint motors used to move the robot arm � We call generalized forces generalized forces the whole set of forces/torques Basilio Bona 2 ROBOTICA 03CFIOR
Statics – 2 τ τ 3 4 τ 5 τ 2 τ 6 TCP τ τ τ τ 1 1 f ⋯ ( ) t τ 2 f ( ) t Ν ( ) t τ def def BASE τ = 3 ⇔ = ( ) t ⋯ F ( ) t τ 4 N ( ) t τ Joint generalized forces 5 τ 6 Cartesian (task space) generalized forces Basilio Bona 3 ROBOTICA 03CFIOR
Statics – 3 T τ = k f � Prismatic joint − − i i 1 i 1, i T τ = k N � Revolute joint i i − 1 i − 1, i τ � To find the relation between F and we use the virtual work principle we use the virtual work principle � TCP generalized forces define a virtual work T δ = F δ W p TCP � Joint generalized forces define another virtual work τ T δ = δ W q g Basilio Bona 4 ROBOTICA 03CFIOR
Statics – 4 � Virtual work principle states that a static equilibrium static equilibrium condition exists when τ T T δ = δ ∀ ⇔ δ = δ W W , q ( ) t q F p g TCP � Virtual displacements are equal to differential δ δ = = δ δ = = q q d , d , q q p p d d p p displacements, i.e., displacements, i.e., � So … = d p J q ( )d q T T τ = d q F J q ( )d q This is the relation between T T T τ = τ = F J J F TCP forces and joint forces. It is an equivalence equivalence relation If one needs to compute the joint T τ = − J F forces needed to equilibrate equilibrate Equilibrate and Balance are synonymous the TCP force, the relation is Basilio Bona 5 ROBOTICA 03CFIOR
Kineto-static duality – 1 ɺ = ɺ p Jq � Since T τ = ± J F we speak of a kineto kineto- -static duality static duality between generalized (cartesian) forces and cartesian velocities. Considering the geometric Jacobian (that has is more geometrically meaningful than the analytical one) we have ɺ = ɺ p J q g T τ = ± J F g � The duality can be characterized considering the range and the kernel of the transformations J T J and g g Basilio Bona 6 ROBOTICA 03CFIOR
Matrix review – 1 Basilio Bona 7 ROBOTICA 03CFIOR
Matrix review – 2 Basilio Bona 8 ROBOTICA 03CFIOR
Matrix review – 3 Basilio Bona 9 ROBOTICA 03CFIOR
Kineto-static duality – 2 T ɺ ɺ = ω = p v J q q ( ) � Consider g ( ) ( ) J q ( ) J q ( ) N � Image space R � Null space g g It contains the TCP velocities that can It contains the joint velocities that do be generated by the joint velocities, for not produce any TCP velocities, for a a given pose given pose T τ = J ( ) q F � Consider g ( ) ( ) T T � Image space � Null space J ( ) q J ( ) q N R g g It contains the joint generalized torques It contains the TCP generalized forces that can balance TCP generalized forces, that do not require balancing joint for a given pose generalized forces , for a given pose Basilio Bona 10 ROBOTICA 03CFIOR
Kineto-static duality – 3 � When the robot is in a singular singular configuration: There are non zero joint velocities There are non zero joint generalized that produce zero TCP velocities forces that cannot be balanced by TCP generalized forces There are TCP generalized forces There are TCP velocities that cannot that do not require any balancing be obtained by any joint velocities joint generalized forces Basilio Bona 11 ROBOTICA 03CFIOR
Conclusions � Statics is important since it allows to compute the equivalent effects on joints of TCP forces (and viceversa) � Statics and velocity kinematics are linked by duality � Remember that the product of a force by a velocity is the power � For this reason they cannot be set at will. If you set a force you cannot set the corresponding velocity and viceversa, since the power is an external constraint � Elastic forces were not considered Basilio Bona 12 ROBOTICA 03CFIOR
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