Statics Basilio Bona DAUIN Politecnico di Torino 2009 Basilio - PowerPoint PPT Presentation

Statics Basilio Bona DAUIN – Politecnico di Torino 2009 Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 1 / 16

Statics - An Introduction Statics studies the equilibrium or equivalence relations between “cartesian” forces and torques, and their counterparts at the robot joints. Cartesian Space The 3 × 1 force and torque vectors f and N     f x N x f = f y N = N y     f z N z acting on the manipulator TCP are collected in a single vector called cartesian generalized force , or briefly cartesian force F � � f F = N Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 2 / 16

Statics - An Introduction Joint Space We call joint generalized forces the vector τ τ 1   . τ = . .   τ n whose elements are the active forces or torques applied by motors to the joints. With active we denote only those components of each force/torque along the motion axis; taking into account the DH conventions, these active components are: τ i k T = i − 1 N i − 1 , i prismatic joint τ i k T = i − 1 f i − 1 , i rotation joint All the other components are acting on the constraints and do no produce work. Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 3 / 16

Virtual Works Principle Work is a real quantity obtained as the vector product of a force/torque and a linear/angular displacement L = N T α L = f T x Taking into account generalized forces, we obtain the “cartesian” work at the TCP L p = F T p and at the joints L j = τ T q Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 4 / 16

Virtual Works Principle The lavoro virtuale is computed considering only the virtual displacements , i.e., those geometrical displacements that are compatible (or admissible) with the constraints acting on the robot structure. The cartesian virtual displacements are called δ p , while the joint virtual displacements are called δ q Hence, the cartesian virtual work will be δ L p = F T δ p while the joint virtual work will be δ L j = τ T δ q Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 5 / 16

Virtual Works Principle The Virtual Works Principle states that in a static equilibrium condition the virtual work of the generalized cartesian forces equals the virtual work of the generalized joint forces δ L p = δ L j τ T δ q = F T δ p ⇔ Recalling the kinematic relation ˙ p = J ˙ q , from its differential equivalent d p = J d q it follows ⇒ δ p = J δ q So we obtain δ L p = δ L j τ T δ q = F T J δ q ⇔ Since the last identity must be true for any δ q , it must follow that τ T = F T J or, transposing the result τ = J T F Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 6 / 16

Kineto-Static Relation We have thus obtained a kineto-static relation τ = J T F that relates the cartesian generalized forces at the TCP with the joint generalized forces. To be precise, the relation should have been written as τ = ± J T F where the ± sign indicates that the work can be assumed positive or negative according to the convention chosen to represent the work done ON the manipulator from the external environment, or the work done BY the manipulator on the external environment. Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 7 / 16

Kineto-Static Relation To synthesize we assume the following conventions τ = J T F is the equivalence relation between τ e F while τ = − J T F is the equilibrium relation between τ e F The first gives the joint generalized forces that are equivalent to the TCP generalized forces. The second gives the value of the same joint generalized forces that are necessary to compensate for the TCP generalized forces. Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 8 / 16

Image Space and Null Space of J and J T The kineto-static relation is particularly important for its implications on velocities and generalized force when the manipulator is in a singularity condition. The kinematic singularity appears when det J = 0. Geometrically, this indicates a rank decrease of the linear transformation q �→ ˙ ˙ p , represented by the jacobian matrix (square for simplicity) J . Rank falls when the null space dimension of J increases from 0 to a positive value. We recall that, given a linear transformation represented by a square matrix M , the image or range space R ( M ) is the space spanned by vectors y , such that y = Mx , x ∈ D ( M ), where D ( M ) is the transformation domain . The null space or null-space or kernel N ( M ) of a linear transformation is the space spanned by vectors x , such that 0 = Mx , x ∈ N ( M ) ⊆ D ( M ). Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 9 / 16

Image Space and Null Space of J and J T Given a transformation M n × n : D ( M ) → R ( M ), the following relation holds n = ρ + ν where ρ = dim R and ν = dim N , if ν = 0 transformation is non singular and M has full rank if 0 < ν ≤ ρ transformation is singular and M decreases its rank many times as ν � = 0. Moreover the following relations are always true for linear transformations between finite spaces R ( M ) = N ( M T ) ⊥ N ( M ) = R ( M T ) ⊥ (1) R ( M T ) = N ( M ) ⊥ N ( M T ) = R ( M ) ⊥ Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 10 / 16

Image Space and Null Space of J and J T If we apply (1) to J and J T , we can establish what follows When J is singular, there are non zero velocities at the joints q s ∈ N ( J ) that produce zero cartesian velocities. ˙ In this case there are moments τ s ∈ N ( J ) that belong to the orthogonal complement of R ( J T ) and that cannot be balanced by any generalized cartesian force F ∈ R ( J ). When J T ` e is singular, we can apply to the TCP a generalized force F s ∈ N ( J T ) that do not requires any joint force to be compensated, i.e., τ s = J T F s = 0 . Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 11 / 16

Example Figure: A planar 2R manipulator. Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 12 / 16

Example The kinematic equations of the 2R manipulator are: x = ℓ 1 c 1 + ℓ 2 c 12    ℓ 1 c 1 + ℓ 2 c 12  y = ℓ 1 s 1 + ℓ 2 s 12 ⇒ p 3 = ℓ 1 s 1 + ℓ 2 s 12   θ z = q 1 + q 2 q 1 + q 2  where p 3 includes the only 3 dof of the plane. Jacobian matrix is − ℓ 1 s 1 − ℓ 2 s 12 − ℓ 2 s 12   � � ℓ 1 c 1 + ℓ 2 c 12 ℓ 2 c 12 J L J =  =   ··· ··· J A  1 1 Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 13 / 16

Example Looking only at the linear part J L , i.e., considering only the linear motion kinematics and assuming for simplicity that ℓ 1 = ℓ 2 = 1 m , we have � � − s 1 − s 12 − s 12 J L = c 1 +c 12 c 12 Computing the determinant we obtain det( J L ) = s 2 The singularity appears when sin( q 2 ) = 0, that is when q 2 = k π , k = 0 , ± 1 , ± 2 ,... In singularity conditions we have that � � � � x ˙ − s 1 = (2˙ q 1 + ˙ q 2 ) y ˙ c 1 Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 14 / 16

Example In singularity, the Jacobian reduces to � � � � − 2s 1 − s 1 − 2s 1 2c 1 J T J s = s = 2c 1 − s 1 c 1 c 1 If we want to study the null-space or the image space, it is easier to assume q 1 = 0; in this way we can easily compute the following sub-spaces � � � � � � � � 0 1 x = λ ; λ ∈ R x = λ ; λ ∈ R R ( J ) = N ( J ) = 1 − 2 � � � � � � � � 2 1 R ( J T ) = x = λ ; λ ∈ R N ( J T ) = x = λ ; λ ∈ R 1 0 Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 15 / 16

Example For example, the joint torque τ T = � 1 − 2 � cannot be balanced by k any generalized force at the TCP, F T = and the generalized force � 1 0 � k does not need any joint bal- ancing torque. Figure: Singularity spaces of the Jacobian matrix. Figure: Which force can balance the two τ i at the same time? Basilio Bona (DAUIN – Politecnico di Torino) Statics 2009 16 / 16