ROBOTICS ROBOTICS 01PEEQW 01PEEQW 01PEEQW 01PEEQW Basilio Bona - PowerPoint PPT Presentation

ROBOTICS ROBOTICS 01PEEQW 01PEEQW 01PEEQW 01PEEQW Basilio Bona Basilio Bona DAUIN DAUIN – – Politecnico di Torino Politecnico di Torino

Statics Statics

Statics – 1 � We call GENERALIZED FORCES GENERALIZED FORCES the whole set of forces and torques � Statics studies the relations between the task space generalized forces (TSGF) and the joint generalized forces (JGF) in static equilibrium conditions � The TSGF derive from possible interactions with the environment (e.g., when the TCP pushes against a surface) � The JGF are provided by the power supplied by the joint motors used to move the robot arms Basilio Bona 3 ROBOTICS 01PEEQW

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 4 ROBOTICS 01PEEQW

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 5 ROBOTICS 01PEEQW

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 “similar” 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 6 ROBOTICS 01PEEQW

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 geometrically more significant than the analytical one) we have ɺ = ɺ p J q g T τ = ± J F g � The duality can be characterized considering the mathematical concepts of range and kernel of the transformations J T J and g g Basilio Bona 7 ROBOTICS 01PEEQW

Matrix review – 1 Basilio Bona 8 ROBOTICS 01PEEQW

Matrix review – 2 Basilio Bona 9 ROBOTICS 01PEEQW

Matrix review – 3 Basilio Bona 10 ROBOTICS 01PEEQW

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 11 ROBOTICS 01PEEQW

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 See Example_2013_02 Basilio Bona 12 ROBOTICS 01PEEQW

Elasticity of the structure � A perfectly rigid robot does not exist in practice � Elastic effects can be localized in 1. Joints, due to the mechanical transmission elements: long motor shafts, belts, chains, gearboxes, etc. 2. Links, due to distributed compliance of the mechanical structure (flexion, torsion, compression) 1 2 Basilio Bona 13 ROBOTICS 01PEEQW

Elasticity – 1 � When a generalized force is applied to the robot TCP a small deflection takes place → ∆ F p � We want to describe the relation in static conditions between the relevant variables τ τ F F , , , , , , p q p q � We introduce an approximated description, considering the elasticity due only to the joints (links are perfectly rigid) Basilio Bona 14 ROBOTICS 01PEEQW

Elasticity – 2 Basilio Bona 15 ROBOTICS 01PEEQW

Elasticity – 3 Basilio Bona 16 ROBOTICS 01PEEQW

Elasticity – 4 Basilio Bona 17 ROBOTICS 01PEEQW

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 a power � For this reason forces and velocities cannot be set at will. � If you set a force you cannot set at will the corresponding velocity and viceversa, since the power is an external constraint � Elastic forces are usually not considered in the robot model, but they are very important in control Basilio Bona 18 ROBOTICS 01PEEQW