Dynamics Basilio Bona DAUIN-Politecnico di Torino 2009 Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30
Dynamics - Introduction In order to determine the dynamics of a manipulator, it is necessary to introduce the force or the inertia concept. The fundamental law of mechanics, established by Newton, is expressed as a vector valued equality f x a x = m f = m a i.e., f y a y f z a z where a = ˙ v is the acceleration and f is the applied force. This relation is applicable to a point mass (or point particle) m , where the applied force f and the acceleration a acting on the particle may alternatively play the role of cause or effect. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 2 / 30
Dynamics - Introduction Similar to the relation f = m a valid for linear motions, there is another one, valid for rotation motions. This one, due to Euler, establishes a relation among the torque applied to a rigid body, its angular velocity and acceleration, and the body inertia moment: τ = Γ ˙ ω + ω × Γ ω where τ is the applied torque, ω and ˙ ω il momento applicato al corpo, sono la velocit` a e l’accelerazione angolare del corpo e Γ ` e il momento d’inerzia del corpo rispetto al suo baricentro. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 3 / 30
Dynamics - Introduction A system composed by N mass points or else a rigid body is in dynamic equilibrium when the sum of all forces, including inertial forces, is zero. the sum of all angular torques (with respect to the body mass center), including inertial torques, is zero. The first condition allows to write differential linear equations of equilibrium, called Newton equations , while the second condition allows to write differential angular equations of equilibrium, called Euler equations . Equations are vectorial, i.e., they relate vectorial quantities; every vectorial equations implies three scalar equations, one for each component. In n links manipulators, the number of vectorial equations is 2 n . Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 4 / 30
Newton-Euler Equations Consider the generic i -th robot arm b i , with its center of mass c i , and DH conventions applied. Figure: Arm i with applied forces and torques. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 5 / 30
Newton Equations If: f i − 1 , i resultant of the forces applied from arm i − 1 to arm i f i +1 , i resultant of the forces applied from arm i +1 to arm i local gravity field acceleration vector g i total acceleration of the center-of-mass a c i We can write the i -th Newton equation as: f i − 1 , i + f i +1 , i + m i g i − m i a c i = 0 Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 6 / 30
Euler Equations If: resultant of the torques applied from arm ( i − 1) to arm i N i − 1 , i N i +1 , i resultant of the forces applied from arm ( i +1) to arm i Γ i inertia matrix of arm i with respect to its center-of-mass r c i , i − 1 position of the ( i − 1) BRF origin with respect to center-of-mass position of the ( i +1) BRF with respect to center-of-mass r c i , i BRF Body Reference Frame We can write the i -th Euler equation as − Γ i ˙ N i − 1 , i + N i +1 , i + r c i , i − 1 × f i − 1 , i + r c i , i × f i +1 , i ω i − ω i × Γ i ω i = 0 � �� � moment of the forces f Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 7 / 30
Newton-Euler Equations The 2 n vectorial Newton-Euler (N-E) equations are difficult to deal with, at least symbolically, since the internal constraints between the arms appear explicitly. These constraints are due to the forces f transmitted by an arm to the other. These constraints have no interest in determining the dynamical behavior and the motion laws of the multi-body structure, and their determination is useless. From this point of view, Lagrange equations are much more immediate and easy to deal with, as we shall see in the following. Nonetheless, from a purely numerical/algorithmic viewpoint N-E equations are easier to solve than Lagrange, due to the several recursive solutions found in literature. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 8 / 30
Lagrange Equations Lagrange equations are scalar differential equations, unlike N-E equations; they are derived from the definition of a Lagrangian function given by the difference of the total kinetic co-energy and the total potential energy of the entire manipulator. Kinetic Co-energy The total kinetic co-energy C is a non-negative scalar function of the joint coordinates q ( t ) and velocities ˙ q ( t ); it is an additive function, i.e., n ∑ C ( q ( t ) , ˙ q ( t )) = C i ( q ( t ) , ˙ q ( t )) i =1 where C i is the kinetic co-energy of arm i . Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 9 / 30
Lagrange Equations Potential Energy In mechanical systems the total potential energy P is a function of the joint coordinates q ( t ) alone; it is additive, i.e., n ∑ P ( q ( t )) = P i ( q ( t )) i =1 where P i is the i -th arm potential energy. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 10 / 30
Lagrange Equations Lagrangian function The Lagrangian (or Lagrangean) function is a state-function defined as n n ∑ ∑ L ( q ( t ) , ˙ q ( t )) = C − P = C i ( q ( t ) , ˙ q ( t )) − P i ( q ( t )) i =1 i =1 “State-function” means that the value of the function depends only on the state of the system at time ( t ). Once the lagrangian is known, the system dynamics is described by n scalar differential equations: � ∂ L � − ∂ L d = F i i = 1 ,..., n d t ∂ ˙ q i ∂ q i where F i is the i -th generalized scalar force . Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 11 / 30
Lagrange Equations From a dimensional point of view, Lagrange equations are differential equations whose terms are scalar forces or torques. Approximately we can say that these forces/torques are the sum of all the effects of the virtual work associated with the i -th coordinate q i ( t ). If among these forces there are also dissipative forces associated to a dissipative function D , where n n q ) = 1 q 2 ∑ ∑ D (˙ q ) = D i (˙ β i ˙ i 2 i =1 i =1 Lagrange equation can be written as � ∂ L � − ∂ L + ∂ D d = F i i = 1 ,..., n d t ∂ ˙ q i ∂ q i ∂ ˙ q i Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 12 / 30
Kinetic Energy Since the energy is an additive scalar quantity, it is convenient to provide only the value of the i -th arm kinetic energy: q ) = 1 q )+ 1 2 v T 2 ω T C i ( q , ˙ c i ( q , ˙ q ) m i v c i ( q , ˙ i ( q , ˙ q ) Γ i ω i ( q , ˙ q ) where v c i is the “total” linear velocity of the i -th body center-of-mass (total means that is the velocity of the body with respect to an inertial RF), where ω i is the total angular velocity of the i -th body and Γ i the inertia matrix with respect to the center-of-mass. Hence kinetic energy of a roto-translating arm is the sum of two terms: the kinetic energy due to translation plus the kinetic energy due to rotation. The former depends on the square value of the center-of-mass velocity, while the latter depends on the square value of the angular velocity of the arm. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 13 / 30
Kinetic Energy To express the kinetic energy (KE) directly in function of the generalized coordinates q and velocities ˙ q it is necessary to know the Jacobian matrices relative to v c i and ω i . If we define � � J ( i ) J ( i ) J ( i ) J ( i ) L = ··· 0 ··· 0 L 1 L 2 Li � � J ( i ) J ( i ) J ( i ) J ( i ) A = ··· ··· 0 0 A 1 A 2 Ai we have � v c i = J ( i ) q 1 + J ( i ) q 2 + ··· + J ( i ) q i = J ( i ) L 1 ˙ L 2 ˙ Li ˙ L ˙ q ⇔ ˙ p c i = J c i ˙ q ω i = J ( i ) q 1 + J ( i ) q 2 + ··· + J ( i ) q i = J ( i ) A 1 ˙ A 2 ˙ Ai ˙ A ˙ q Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 14 / 30
Kinetic Energy We can therefore write the KE as 1 q T � � q T � � L ( q ) T m i J ( i ) A ( q ) T Γ i J ( i ) J ( i ) J ( i ) 2 ∑ ˙ L ( q ) q + ˙ ˙ A ( q ) ˙ q i or else 1 � � � � L ( q ) T m i J ( i ) A ( q ) T Γ i J ( i ) J ( i ) J ( i ) q T ∑ 2 ˙ L ( q ) + A ( q ) ˙ q i � �� � H i ( q ) and C = 1 q = 1 q T ∑ q T H ( q )˙ 2 ˙ ( H i ( q )) ˙ 2 ˙ q i Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 15 / 30
Kinetic Energy Calling H ij ( q ) the generic term ij of matrix H ( q ), we observe that n ∑ H ij ( q ) = { H ij ( q ) } k k =1 where { H ij ( q ) } k is the ij -th element of the inertia matrix of the k -th arm. Therefore, another formulation of the kinetic co-energy is the following: n n C = 1 ∑ ∑ H ij ( q )˙ q i ˙ q j 2 i =1 j =1 Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 16 / 30
Potential Energy Usually potential energy is stored in elastic elements; if the manipulator is composed of purely rigid arms, there are no elements capable to store potential energy. Nonetheless, we can also consider another form of kinetic energy, i.e., the “position energy”, due to the presence of a gravitational force field, that, locally, has field lines parallel to the local acceleration vector G . Figure: Potential energy has equi-potential surfaces orthogonal to the local gravity vector G . Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 17 / 30
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