CHAPTER 6
DYNAMICS

6.1.1 Motivation

To motivate the subsequent discussion, we show first how the Euler–Lagrange equations can be derived from Newton’s second law for the one-degree-of-freedom system shown in Figure 6.1.

By Newton’s second law, the equation of motion of the particle is

(6.1)

Notice that the left-hand side of Equation (6.1) can be written as

(6.2)

where is the kinetic energy. We use the partial derivative notation in the above expression to be consistent with systems considered later when the kinetic energy will be a function of several variables. Likewise we can express the gravitational force in Equation (6.1) as

(6.3)

where is the potential energy due to gravity. If we define

(6.4)

and note that

then we can write Equation (6.1) as

(6.5)

The function , which is the difference of the kinetic and potential energy, is called the Lagrangian of the system, and Equation (6.5) is called the Euler–Lagrange equation.

The general procedure that we discuss below is, of course, the reverse of the above; namely, one first writes the kinetic and potential energies of a system in terms of a set of so-called generalized coordinates (q₁, …, q_n), where n is the number of degrees of freedom of the system, and then computes the equations of motion of the n-DOF system according to

(6.6)

where τ_k is the (generalized) force associated with q_k. In the above single DOF example, the variable y serves as the generalized coordinate. Application of the Euler–Lagrange equations leads to a set of coupled second-order ordinary differential equations and provides a formulation of the dynamic equations of motion for serial-link robots equivalent to those derived using Newton’s second law. However, as we shall see, the Lagrangian approach is advantageous for complex systems such as multi-link robots.

Example 6.1. (Single-Link Manipulator).

(6.7)

(6.8)

(6.9)

(6.10)

(6.11)

Example 6.2. (Single-Link Manipulator with Elastic Joint).

(6.12)

(6.13)

(6.14)

6.1.2 Holonomic Constraints and Virtual Work

Now, consider a system of k particles with corresponding position vectors r₁, …, r_k as shown in Figure 6.4.

If these particles are free to move about without any restrictions, then it is quite an easy matter to describe their motion, by noting that the mass times acceleration of each particle equals the external force applied to it. However, if the motion of the particles is constrained in some fashion, then one must take into account not only the externally applied forces, but also the so-called constraint forces, that is, the forces needed to make the constraints hold. As a simple illustration of this, suppose the system consists of two particles joined by a massless rigid wire of length ℓ. Then the two coordinates r₁ and r₂ must satisfy the constraint

(6.15)

If one applies some external forces to each particle, then the particles experience not only these external forces but also the force exerted by the wire, which is along the direction r₂ − r₁ and of appropriate magnitude. Therefore, in order to analyze the motion of the two particles, we could follow one of two options. We could compute, under each set of external forces, what the corresponding constraint force must be in order that the equation above continues to hold. Alternatively, we can search for a method of analysis that does not require us to know the constraint force. Clearly, the second alternative is preferable, since it is generally quite an involved task to compute the constraint forces. This section is aimed at achieving this latter objective.

First, it is necessary to introduce some terminology. A constraint on the k coordinates r₁, …, r_k is called holonomic if it is an equality constraint of the form

(6.16)

The constraint given in Equation (6.15) imposed by connecting two particles by a massless rigid wire is an example of a holonomic constraint. By differentiating Equation (6.16) we have an expression of the form

(6.17)

A constraint of the form

(6.18)

is called nonholonomic if it cannot be integrated to an equality constraint of the form (6.16). It is interesting to note that, while the method of deriving the equations of motion using the principle of virtual work remains valid for nonholonomic systems, methods based on variational principles, such as Hamilton’s principle, can no longer be applied to derive the equations of motion. We will discuss systems subject to nonholonomic constraints in Chapter 14.

If a system is subjected to ℓ holonomic constraints, then one can think in terms of the constrained system having ℓ fewer degrees of freedom than the unconstrained system. In this case, it may be possible to express the coordinates of the k particles in terms of n generalized coordinates q₁, …, q_n. In other words, we assume that the coordinates of the various particles, subjected to the set of constraints given by Equation (6.16), can be expressed in the form

(6.19)

where q₁, …, q_n are all independent. In fact, the idea of generalized coordinates can be used even when there are infinitely many particles. For example, a physical rigid object such as a bar contains an infinity of particles; but since the distance between each pair of particles is fixed throughout the motion of the bar, six coordinates are sufficient to specify completely the coordinates of any particle in the bar. In particular, one could use three position coordinates to specify the location of the center of mass of the bar, and three Euler angles to specify the orientation of the body. Typically, generalized coordinates are positions, angles, etc. In fact, in Chapter 3 we chose to denote the joint variables by the symbols q₁, …, q_n precisely because these joint variables form a set of generalized coordinates for an n-link robot manipulator.

One can now speak of virtual displacements, which are any set of infinitesimal displacements, δr₁, …, δr_k, that are consistent with the constraints. For example, consider once again the constraint (6.15) and suppose r₁, r₂ are perturbed to r₁ + δr₁, r₂ + δr₂, respectively. Then, in order that the perturbed coordinates continue to satisfy the constraint, the length of the bar must not change and so we must have

(6.20)

Now, let us expand the above product and take advantage of the fact that the original coordinates r₁, r₂ satisfy the constraint given by Equation (6.15). If we neglect quadratic terms in δr₁, δr₂ we obtain after some algebra (Problem 6–2)

(6.21)

Thus, any infinitesimal perturbations in the positions of the two particles must satisfy the above equation in order that the perturbed positions continue to satisfy the constraint Equation (6.15). Any pair of infinitesimal vectors δr₁, δr₂ that satisfy Equation (6.21) constitutes a set of virtual displacements for this problem. Figure 6.5 shows some representative virtual displacements for a rigid bar.

Now, the reason for using generalized coordinates is to avoid dealing with complicated relationships such as Equation (6.21) above. If Equation (6.19) holds, then one can see that the set of all virtual displacements is precisely

(6.22)

where the virtual displacements δq₁, …, δq_n of the generalized coordinates are unconstrained (that is what makes them generalized coordinates).

Next, we begin a discussion of constrained systems in equilibrium. Suppose each particle is in equilibrium. Then the net force on each particle is zero, which in turn implies that the work done by each set of virtual displacements is zero. Hence, the sum of the work done by any set of virtual displacements is also zero; that is,

(6.23)

where F_i is the total force on particle i. As mentioned earlier, the force F_i is the sum of two quantities, namely (i) the externally applied force f_i, and (ii) the constraint force f^a_i. Now, suppose that the total work done by the constraint forces corresponding to any set of virtual displacements is zero, that is,

(6.24)

This will be true whenever the constraint force between a pair of particles is directed along the radial vector connecting the two particles (see the discussion in the next paragraph). Substituting Equation (6.24) into Equation (6.23) results in

(6.25)

The beauty of this equation is that it does not involve the unknown constraint forces, but only the known external forces. This equation expresses the principle of virtual work, which can be stated in words as: The work done by external forces corresponding to any set of virtual displacements is zero.

Note that the principle is not universally applicable; it requires that Equation (6.24) hold, that is, that the constraint forces do no work. Thus, if the principle of virtual work applies, one can analyze the dynamics of a system without having to evaluate the constraint forces.

It is easy to verify that the principle of virtual work applies whenever the constraint force between a pair of particles acts along the vector connecting the position coordinates of the two particles. In particular, when the constraints are of the form (6.15), the principle applies. To see this, consider once again a single constraint of the form (6.15). In this case, the constraint force, if any, must be exerted by the rigid massless wire, and therefore must be directed along the radial vector connecting the two particles. In other words, the force exerted on the first particle by the wire must be of the form

(6.26)

for some constant c (which could change as the particles move about). By the law of action and reaction, the force exerted on the second particle by the wire must be just the negative of the above, that is,

(6.27)

Now, the work done by the constraint forces corresponding to a set of virtual displacements is

(6.28)

But, Equation (6.21) shows that the above expression must be zero for any set of virtual displacements. The same reasoning can be applied if the system consists of several particles that are pairwise connected by rigid massless wires of fixed lengths, in which case the system is subjected to several constraints of the form (6.15). Now, the requirement that the motion of a body be rigid can be equivalently expressed as the requirement that the distance between any pair of points on the body remain constant as the body moves, that is, as an infinity of constraints of the form (6.15). Thus, the principle of virtual work applies whenever rigidity is the only constraint on the motion. There are indeed situations when this principle does not apply, such as in the presence of magnetic fields. However, in all situations encountered in this book, we can safely assume that the principle of virtual work is valid.

6.1.3 D’Alembert’s Principle

In Equation (6.25), the virtual displacements δr_i are not independent, so we cannot conclude from this equation that each coefficient F_i individually equals zero. In order to apply such reasoning, we must transform to generalized coordinates. Before doing this, we consider systems that are not necessarily in equilibrium. For such systems, D’Alembert’s principle states that, if one introduces a fictitious additional force on each particle, where p_i is the momentum of particle i, then each particle will be in equilibrium. Thus, if one modifies Equation (6.23) by replacing F_i by , then the resulting equation is valid for arbitrary systems. One can then remove the constraint forces as before using the principle of virtual work. This results in the equation

(6.29)

The above equation does not mean that each coefficient of δr_i is zero since the virtual constraints δr_i are not independent. The remainder of this derivation is aimed at expressing the above equation in terms of the generalized coordinates, which are independent. For this purpose, we express each δr_i in terms of the corresponding virtual displacements of generalized coordinates, as is done in Equation (6.22). Then, the virtual work done by the forces f_i is given by

(6.30)

where

(6.31)

is called the j^th generalized force. Note that ψ_j need not have dimensions of force, just as q_j need not have dimensions of length; however, ψ_jδq_j must always have dimensions of work.

Now, let us study the second summation in Equation (6.29). Since , it follows that

(6.32)

Next, using the product rule of differentiation, we have

(6.33)

Rearranging the above and summing over all i = 1, …, n yields

(6.34)

Now, differentiating Equation (6.19) using the chain rule gives

(6.35)

Observe from the above equation that

(6.36)

Next,

(6.37)

where the last equality follows from Equation (6.35). Substituting from Equation (6.36) and Equation (6.37) into Equation (6.34) and noting that gives

(6.38)

If we define the kinetic energy K to be the quantity

(6.39)

then Equation (6.38) can be compactly expressed as

(6.40)

Now, substituting from Equation (6.40) into Equation (6.32) shows that the second summation in Equation (6.29) is

(6.41)

Finally, combining Equations (6.29), (6.30), and (6.41) gives

(6.42)

Now, since the virtual displacements δq_j are independent, we can conclude that each coefficient in Equation (6.42) is zero, that is,

(6.43)

If the generalized force ψ_j is the sum of an externally applied generalized force and another one due to a potential field, then a further modification is possible. Suppose there exist functions τ_j and a potential energy function P(q) such that

(6.44)

Then Equation (6.43) can be written in the form

(6.45)

where is the Lagrangian and we have recovered the Euler–Lagrange equations of motion as in Equation (6.6).

6.2.1 The Inertia Tensor

It is understood that the above linear and angular velocity vectors, v and ω, respectively, are expressed in the inertial frame. In this case, we know that ω is found from the skew-symmetric matrix

(6.47)

where R is the orientation transformation from the body-attached frame and the inertial frame. It is therefore necessary to express the inertia tensor, , also in the inertial frame in order to compute the triple product . The inertia tensor relative to the inertial reference frame will depend on the configuration of the object. If we denote as I the inertia tensor expressed instead in the body-attached frame, then the two matrices are related via a similarity transformation according to

(6.48)

This is an important observation because the inertia matrix expressed in the body-attached frame is a constant matrix independent of the motion of the object and easily computed.

We next show how to compute this matrix explicitly. Let the mass density of the object be represented as a function of position, ρ(x, y, z). Then the inertia tensor in the body attached frame is computed as

(6.49)

where

and

The integrals in the above expression are computed over the region of space occupied by the rigid body. The diagonal elements of the inertia tensor, I_xx, I_yy, I_zz, are called the principal moments of inertia about the x, y, and z-axes, respectively. The off-diagonal terms I_xy, I_xz, etc., are called the cross products of inertia. If the mass distribution of the body is symmetric with respect to the body-attached frame, then the cross products of inertia are identically zero.

Example 6.3. (Uniform Rectangular Solid).

Consider the rectangular solid of length a width b and height c shown in Figure 6.7 and suppose that the density is constant, ρ(x, y, z) = ρ. If the body frame is attached at the geometric center of the object, then by symmetry, the cross products of inertia are all zero and it is a simple exercise to compute

since ρabc = m, the total mass. Likewise, a similar calculation shows that

6.2.2 Kinetic Energy for an n-Link Robot

Now, consider a manipulator with n links. We have seen in Chapter 4 that the linear and angular velocities of any point on any link can be expressed in terms of the Jacobian matrix and the derivatives of the joint variables. Since, in our case, the joint variables are indeed generalized coordinates, it follows that, for appropriate Jacobian matrices and , of dimension 3 × n, we have

(6.50)

Now, suppose the mass of link i is m_i and that the inertia matrix of link i, evaluated around a coordinate frame parallel to frame i but whose origin is at the center of mass, equals I_i. Then from Equations (6.46) and (6.50) it follows that the overall kinetic energy of the manipulator equals

(6.51)

(6.52)

where

(6.53)

is an n × n configuration dependent matrix called the inertia matrix. In Section 6.4 we will compute this matrix for several commonly occurring manipulator configurations. The inertia matrix is symmetric and positive definite for any manipulator. Symmetry of D(q) is easily seen from Equation (6.53). Positive definiteness can be inferred from the fact that the kinetic energy is always nonnegative and is zero if and only if all of the joint velocities are zero. The formal proof is left as an exercise (Problem 6–6).

6.2.3 Potential Energy for an n-Link Robot

Now, consider the potential energy term. In the case of rigid dynamics, the only source of potential energy is gravity. The potential energy of the i^th link can be computed by assuming that the mass of the entire object is concentrated at its center of mass and is given by

(6.54)

where g is the vector giving the direction of gravity in the inertial frame and the vector r_ci gives the coordinates of the center of mass of link i. The total potential energy of the n-link robot is therefore

(6.55)

In the case that the robot contains elasticity, for example if the joints are flexible, then the potential energy will include terms containing the energy stored in the elastic elements. Note that the potential energy is a function only of the generalized coordinates and not their derivatives.

Two-Link Cartesian Manipulator

Consider the manipulator shown in Figure 6.8 consisting of two links and two prismatic joints. Denote the masses of the two links by m₁ and m₂, respectively, and denote the displacement of the two prismatic joints by q₁ and q₂, respectively. It is easy to see, as mentioned in Section 6.1, that these two quantities serve as generalized coordinates for the manipulator. Since the generalized coordinates have dimensions of distance, the corresponding generalized forces have units of force. In fact, they are just the forces applied at each joint. Let us denote these by f_i, i = 1, 2.

Since we are using the joint variables as the generalized coordinates, we know that the kinetic energy is of the form (6.56) and that the potential energy is only a function of q₁ and q₂. Hence, we can use the formulas in Section 6.3 to obtain the dynamical equations. Also, since both joints are prismatic, the angular velocity Jacobian is zero and the kinetic energy of each link consists solely of the translational term.

It follows that the velocity of the center of mass of link 1 is given by

(6.69)

where

(6.70)

Similarly,

(6.71)

where

(6.72)

Hence, the kinetic energy is given by

(6.73)

Comparing with Equation (6.56), we see that the inertia matrix D is given simply by

(6.74)

Next, the potential energy of link 1 is m₁gq₁, while that of link 2 is m₂gq₁, where g is the acceleration due to gravity. Hence, the overall potential energy is

(6.75)

Now, we are ready to write down the equations of motion. Since the inertia matrix is constant, all Christoffel symbols are zero. Furthermore, the components g_k of the gravity vector are given by

(6.76)

Substituting into Equation (6.65) gives the dynamical equations as

(6.77)

Planar Elbow Manipulator

Now, consider the planar manipulator with two revolute joints shown in Figure 6.9. Let us fix notation as follows: For i = 1, 2, q_i denotes the joint angle, which also serves as a generalized coordinate; m_i denotes the mass of link i; ℓ_i denotes the length of link i; ℓ_ci denotes the distance from the previous joint to the center of mass of link i; and I_i denotes the moment of inertia of link i about an axis coming out of the page, passing through the center of mass of link i.

We will use the Denavit–Hartenberg joint variables as generalized coordinates, which will allow us to make effective use of the Jacobian expressions in Chapter 4 in computing the kinetic energy. First,

(6.78)

where,

(6.79)

Similarly,

(6.80)

where

(6.81)

Hence, the translational part of the kinetic energy is

(6.82)

Next, we consider the angular velocity terms. Because of the particularly simple nature of this manipulator, many of the potential difficulties do not arise. First, it is clear that

(6.83)

when expressed in the base inertial frame. Moreover, since ω_i is aligned with the z-axes of each joint coordinate frame, the rotational kinetic energy reduces simply to , where I_i is the moment of inertia about an axis through the center of mass of link i parallel to the z_i-axis. Hence, the rotational kinetic energy of the overall system in terms of the generalized coordinates is

(6.84)

Now, we are ready to form the inertia matrix D(q). For this purpose, we merely have to add the two matrices in Equation (6.82) and Equation (6.84), respectively. Thus

(6.85)

Carrying out the above multiplications and using the standard trigonometric identities cos ²θ + sin ²θ = 1, cos αcos β + sin αsin β = cos (α − β) leads to

(6.86)

Now, we can compute the Christoffel symbols using Equation (6.63). This gives

Next, the potential energy of the manipulator is just the sum of those of the two links. For each link, the potential energy is just its mass multiplied by the gravitational acceleration and the height of its center of mass. Thus

and so the total potential energy is

(6.87)

Therefore, the functions g_k defined in Equation (6.64) become

(6.88)

(6.89)

Finally, we can write down the dynamical equations of the system as in Equation (6.65). Substituting for the various quantities in this equation and omitting zero terms leads to

(6.90)

In this case, the matrix is given as

(6.91)

Planar Elbow Manipulator with Remotely Driven Link

Now, we illustrate the use of Lagrangian equations in a situation where the generalized coordinates are not the joint variables defined in earlier chapters. Consider again the planar elbow manipulator, but suppose now that both joints are driven by motors mounted at the base. The first joint is turned directly by one of the motors, while the other is turned via a gearing mechanism or a timing belt (see Figure 6.10).

In this case, one should choose the generalized coordinates as shown in Figure 6.11, because the angle p₂ is determined by driving motor number 2, and is not affected by the angle p₁. We will derive the dynamical equations for this configuration and show that some simplifications will result.

Since p₁ and p₂ are not the joint angles used earlier, we cannot use the velocity Jacobians derived in Chapter 4 in order to find the kinetic energy of each link. Instead, we have to carry out the analysis directly. It is easy to see that

(6.92)

(6.93)

(6.94)

Hence, the kinetic energy of the manipulator equals

(6.95)

where

(6.96)

Computing the Christoffel symbols as in Equation (6.63) gives

(6.97)

Next, the potential energy of the manipulator, in terms of p₁ and p₂, equals

(6.98)

Hence, the gravitational generalized forces are

Finally, the equations of motion are

(6.99)

Comparing Equation (6.99) and Equation (6.90), we see that by driving the second joint remotely from the base we have eliminated the Coriolis forces, but we still have the centrifugal forces coupling the two joints.

Five-Bar Linkage

Now, consider the manipulator shown in Figure 6.12. We will show that, if the parameters of the manipulator satisfy a simple relationship, then the equations of the manipulator are decoupled, so that each quantity q₁ and q₂ can be controlled independently of the other. The mechanism in Figure 6.12 is called a five-bar linkage. Clearly, there are only four bars in the figure, but in the theory of mechanisms it is a convention to count the ground as an additional linkage, which explains the terminology. It is assumed that the lengths of links 1 and 3 are the same, and that the two lengths marked ℓ₂ are the same; in this way the closed path in the figure is in fact a parallelogram, which greatly simplifies the computations. Notice, however, that the quantities ℓ_c1 and ℓ_c3 need not be equal. For example, even though links 1 and 3 have the same length, they need not have the same mass distribution. It is clear from the figure that, even though there are four links being moved, there are in fact only two degrees of freedom, identified as q₁ and q₂. Thus, in contrast to the earlier mechanisms studied in this book, this one is a closed kinematic chain (though of a particularly simple kind). As a result, we cannot use the earlier results on Jacobian matrices, and instead have to start from scratch. As a first step we write down the coordinates of the centers of mass of the various links as a function of the generalized coordinates. This gives

(6.100)

(6.101)

(6.102)

(6.103)

Next, with the aid of these expressions, we can write down the velocities of the various centers of mass as a function of and . For convenience we drop the third row of each of the following Jacobian matrices as it is always zero. The result is

(6.104)

Let us define the velocity Jacobians , i ∈ {1, …, 4} in the obvious fashion, that is, as the four matrices appearing in the above equations. Next, it is clear that the angular velocities of the four links are simply given by

(6.105)

Thus, the inertia matrix is given by

(6.106)

If we now substitute from Equation (6.104) into the above equation and use the standard trigonometric identities, we are left with

(6.107)

Now, we note from the above expressions that if

(6.108)

then d₁₂ and d₂₁ are zero, that is, the inertia matrix is diagonal and constant. As a consequence the dynamical equations will contain neither Coriolis nor centrifugal terms.

Turning now to the potential energy, we have that

(6.109)

Hence,

(6.110)

Notice that g₁ depends only on q₁ but not on q₂ and similarly that g₂ depends only on q₂ but not on q₁. Hence, if the relationship (6.108) is satisfied, then the rather complex-looking manipulator in Figure 6.12 is described by the decoupled set of equations

(6.111)

This discussion helps to explain the popularity of the parallelogram configuration in industrial robots. If the relationship (6.108) is satisfied, then one can adjust the two angles q₁ and q₂ independently, without worrying about interactions between the two angles. Compare this with the situation in the case of the planar elbow manipulators discussed earlier in this section.

6.5.1 Skew Symmetry and Passivity

The skew symmetry property refers to an important relationship between the inertia matrix D(q) and the matrix appearing in Equation (6.66).

Proposition 6.1 (The Skew Symmetry Property).

Proof: Given the inertia matrix D(q), the (k, j)^th component of is given by the chain rule as

(6.112)

Therefore, the (k, j)^th component of is given by

(6.113)

Since the inertia matrix D(q) is symmetric, that is, d_ij = d_ji, it follows from Equation (6.113) by interchanging the indices k and j that

(6.114)

which completes the proof.

It is important to note that, in order for to be skew-symmetric, one must define C according to Equation (6.67). This will be important in later chapters when we discuss robust and adaptive control algorithms.

Related to the skew symmetry property is the so-called passivity property which, in the present context, means that there exists a constant, β ⩾ 0, such that

(6.115)

The term has units of power. Thus, the expression is the energy produced by the system over the time interval [0, T]. Passivity means that the amount of energy dissipated by the system has a lower bound given by − β. The word passivity comes from circuit theory where a passive system according to the above definition is one that can be built from passive components (resistors, capacitors, inductors). Likewise a passive mechanical system can be built from masses, springs, and dampers.

To prove the passivity property, let H be the total energy of the system, that is, the sum of the kinetic and potential energies,

(6.116)

The derivative satisfies

(6.117)

where we have substituted for using the equations of motion. Collecting terms and using the fact that yields

(6.118)

the latter equality following from the skew symmetry property. Integrating both sides of Equation (6.118) with respect to time gives,

(6.119)

since the total energy H(T) is nonnegative, and the passivity property therefore follows with β = H(0).

6.5.2 Bounds on the Inertia Matrix

We have remarked previously that the inertia matrix for an n-link rigid robot is symmetric and positive definite. For a fixed value of the generalized coordinate q, let 0 < λ₁(q) ⩽ … ⩽ λ_n(q) denote the n eigenvalues of D(q). These eigenvalues are positive as a consequence of the positive definiteness of D(q). As a result, it can easily be shown that

(6.120)

where I_{n × n} denotes the n × n identity matrix. The above inequalities are interpreted in the standard sense of matrix inequalities, namely, if A and B are n × n matrices, then B < A means that the matrix A − B is positive definite and B ⩽ A means that A − B is positive semi-definite.

If all of the joints are revolute, then the inertia matrix contains only terms involving sine and cosine functions and, hence, is bounded as a function of the generalized coordinates. As a result, one can find constants λ_m and λ_M that provide uniform (independent of q) bounds in the inertia matrix

(6.121)

6.5.3 Linearity in the Parameters

The robot equations of motion are defined in terms of certain parameters, such as link masses, moments of inertia, etc., that must be determined for each particular robot in order, for example, to simulate the equations or to tune controllers. The complexity of the dynamic equations makes the determination of these parameters a difficult task. Fortunately, the equations of motion are linear in these inertia parameters in the following sense. There exists an n × ℓ function, and an ℓ-dimensional vector Θ such that the Euler–Lagrange equations can be written as

(6.122)

The function is called the regressor and is the parameter vector. The dimension of the parameter space, that is, the number of parameters needed to write the dynamics in this way, is not unique. In general, a given rigid body is described by ten parameters, namely, the total mass, the six independent entries of the inertia tensor, and the three coordinates of the center of mass. An n-link robot then has a maximum of 10n dynamics parameters. However, since the link motions are constrained and coupled by the joint interconnections, there are actually fewer than 10n independent parameters. Finding a minimal set of parameters that can parameterize the dynamic equations is, however, difficult in general.

Consider the two-link, revolute-joint, planar robot from Section 6.4. If we group the inertia terms appearing in Equation (6.86) as

(6.123)

(6.124)

(6.125)

then we can write the inertia matrix elements as

(6.126)

(6.127)

(6.128)

No additional parameters are required in the Christoffel symbols as these are functions of the elements of the inertia matrix. However, the gravitational torques generally require additional parameters. Setting

(6.129)

(6.130)

we can write the gravitational terms g₁ and g₂ as

(6.131)

(6.132)

Substituting these into the equations of motion it is straightforward to write the dynamics in the form (6.122) where

and the parameter vector Θ is given by

(6.133)

Thus, we have parameterized the dynamics using a five dimensional parameter space. Note that in the absence of gravity only three parameters are needed.

6.6.1 Planar Elbow Manipulator Revisited

In this section we apply the recursive Newton–Euler formulation derived in Section 6.6 to analyze the dynamics of the planar elbow manipulator of Figure 6.9, and show that the Newton–Euler method leads to the same equations as the Lagrangian method, namely Equation (6.90).

We begin with the forward recursion to express the various velocities and accelerations in terms of q₁, q₂, and their derivatives. Note that, in this simple case, it is quite easy to see that

(6.164)

so that there is no need to use Equations (6.151) and (6.155). Also, the vectors that are independent of the configuration are as follows:

(6.165)

(6.166)

where i here denotes the unit vector (1, 0, 0) and not the index i in the iteration.

Forward Recursion Link 1

Using Equation (6.160) with index i = 1 and noting that a_{e, 0} = 0 gives

(6.167)

where, again i and j the standard unit vectors in the x and y directions, respectively. Notice how simple this computation is when we do it with respect to frame 1, as opposed to the same computation in frame 0. Finally, we have

(6.168)

where g is the acceleration due to gravity. At this stage we can economize a bit by not displaying the third components of these accelerations, since they are all equal to zero. Similarly, the third component of all forces will be zero while the first two components of all torques will be zero. To complete the computations for link 1, we compute the acceleration of end of link 1. This is obtained from Equation (6.167) by replacing ℓ_c1 by ℓ₁. Thus

(6.169)

Forward Recursion: Link 2

Once again we use Equation (6.160) and substitute for ω₂ from Equation (6.164). This yields

(6.170)

The only quantity in the above equation which is configuration dependent is the first one. This can be computed as

(6.171)

Substituting into Equation (6.170) gives

(6.172)

The gravitational vector is

(6.173)

Since there are only two links, there is no need to compute a_{e, 2}. Hence the forward recursions are complete at this point.

Backward Recursion: Link 2

Now we carry out the backward recursion to compute the forces and joint torques. Note that, in this instance, the joint torques are the externally applied quantities, and our ultimate objective is to derive dynamical equations involving the joint torques. First we apply Equation (6.148) with index i = 2 and note that f₃ = 0. This results in

(6.174)

(6.175)

Now we can substitute for ω₂, α₂ from Equation (6.164), and for a_{c, 2} from Equation (6.172). We also note that the gyroscopic term equals zero, since both ω₂ and I₂ω₂ are aligned with k. Now the cross product f₂ × ℓ_c2i is clearly aligned with k, the unit vector in the z direction, and its magnitude is just the second component of f₂. The final result is

(6.176)

Since τ₂ = τ₂k, we see that the above equation is the same as the second equation in (6.90).

Backward Recursion: Link 1

To complete the derivation, we apply Equations (6.148) and (6.149) with index i = 1. First, the force equation is

(6.177)

and the torque equation is

(6.178)

Now we can simplify things a bit. First, R¹₂τ₂ = τ₂, since the rotation matrix does not affect the third components of the vectors. Second, the gyroscopic term is again equal to zero. Finally, when we substitute for f₁ from Equation (6.177) into Equation (6.178), a little algebra gives

(6.179)

Once again, all these products are quite straightforward, and the only difficult calculation is that of R¹₂f₂. The final result is:

(6.180)

If we now substitute for τ₁ from Equation (6.176) and collect terms, we will get the first equation in (6.90); the details are routine and are left to the reader.