Riemannian geometry is in many ways fundamental to robotics. Robotic configuration spaces are most naturally modeled as a manifold , the classical mechanical dynamics of the robot is intimately linked to a Riemannian geometry 
, and natural gradient methods in machine learning. Riemannian geometry has made an impact on planning[12, 15].
However, Riemannian geometry, as a result of its obscure tensor notation, coordinate free descriptions, and focus on affine connections and curvature, remains exceedingly challenging to the general robotics audience. And importantly, there exists natural generalizations Riemannian geometry, such as spray and Finsler geometries, with undoubtably numerous impactful applications, that remain largely untouched by the community. If Riemannian geometry is challenging, these broader nonlinear geometries are impenetrable.
In studying the theoretical foundations of a modern line of reactive motion generation research, exemplified by Riemannian Motion Policies (RMPs)  and the RMPflow algorithm , we recently found ourselves in need of understanding the fundamentals of generalized nonlinear geometry and Finsler geometry. Existing texts are extremely challenging, and we ultimately re-derived the core results rigorously from scratch, building from familiar concepts of advanced calculus  and the Calculus of Variations . These results led to the development of a strong theoretically sound robust generalization of (RMPs) we call Geometric Fabrics , which inheret the intuitive design and modularity of RMPs while attaining stability and a form of geometric consistency that RMPs lacked. Geometric fabrics were a major advance in our understanding of modular reactive policies of this sort, and they have given us a powerful practical toolset for design. And none of that work would have been possible without a core understanding of these generalized nonlinear and Finsler geometries.
In this paper, we present our re-derivation of generalized nonlinear and Finsler geometry, providing intuition wherever possible. It is our hope that this work will inspire and enable many new applications of these geometries in robotics well beyond our initial exploration into geometric fabrics. Our contributions are:
We develop a class of generalized nonlinear geometries of paths characterized by homogeneous differential equations, focusing on their most fundamental form and their defining geometric path consistency properties.
We develop Finsler geometry from a Calculus of Variations perspective using notation familiar from advanced calculus and robotics (classical mechanics), using Riemannian geometry as an early example.
We prove the formal connection between dynamical systems derived from Finsler energies (including standard kinetic energy) and the corresponding nonlinear geometries of paths. This result extends the classical result of geometric mechanics connecting classical mechanics to Riemannian geometry.
Beyond generalized geometries, our derivations make Riemannian geometry, itself more accessible, and place it within the broader context of Finsler geometry.
Despite our novel treatment, this material is still challenging. It is our hope, however, that this presentation is at least accessible to many readers in ways the original mathematical material isn’t. We hope this work bridges these powerful tools to a community of practitioners who will undoubtedly find important and influential applications.
I-a A note on the literature
Many mathematical presentations of the ideas developed here exist (see  for a good introduction to Finsler geometry and  for an in-depth presentation of the generalization to spray geometry). However, the combination of abstract coordinate-free presentation and tensor notation for calculations, as well as their focus on highly theoretical ideas such as general nonlinear connections and global properties of curvature, make them highly inaccessible to most readers. We have found that practically much of the material around curvature is not used in robotic applications,111This is not to say it never will be—it’s fundamental to applications in theoretical physics, especially in general relativity. A clear and accessible treatment of curvature would be in order should we find critical applications in robotics as well. These notions of curvature are more tricking in non-Riemannian geometry anyhow. where path consistency and modeling of the resulting geodesic differential equations tend too be much more applicable (similar in areas of optimization on manifolds, such as those described in ). We, therefore, focus on deriving primarily the pre-curvature theory that relates more strongly to modeling differential equation behavior.
In the mathematical literature, generalizations of nonlinear geometries beyond Finsler tend to focus on what are call sprays and semi-sprays. We find these terms can be confusing for many readers as term spray pertains to more general differential equations with less structure and the term semi-spray pertains to the special class with more geometric path consistency.222In mathematics, nonlinear connections and various definitions of curvature can be derived for a very broad class of second-order differential equation, so they call such differential equations sprays (the geometries can be characterized by their trajectory behavior at any point for the set of all initial valued problems defined by the tangent space at that point). A semi-spray, therefore, is just a particular subclass that is more geometrically consistent (trajectories pointing in the same direction follow the same path independent of initial speed). We, therefore, avoid these terms altogether, and call the more general class of geometries we derive in Section II generalized nonlinear geometries and focus solely on the case where geometries can be characterized based on their speed-independent path consistency (we sometimes refer to these as geometries of paths). This generalization fits well with the fundamental path-consistency properties of Finsler geometries (see Section III).
I-B Notation and manifold concepts
In order to best serve a broad community of readers, we do not dwell on standard notions or definitions of manifolds and covariance. We instead follow what is done commonly in presentations of classical mechanics  and analytical dynamics  which is to assume we are working in a particular coordinate system. For those familiar with ideas around covariance [8, 4] we point out that the Euler-Lagrange equation which is used to derive many of the equations below is fundamentally covariant, which means practically that we can change coordinates as necessary to any curvilinear coordinate system and the fundamental behavior remains the same. The use of transform trees in systems modeling can also be used to attain covariance when the underlying equations themselves are not .
We avoid tensor notation altogether and instead derive all of our expressions using the notation of advanced calculus sticking to vector and matrix notations familiar to roboticists. This notation works well for our purposes since we avoid explicit discussions of curvature and connections, which are of interest primarily to mathematicians and theoretical physicists.
Ii Generalized nonlinear geometries
A generalized nonlinear geometry is a second-order differential equation describing a smooth collection of paths. These paths are equivalence classes of trajectories all passing through the same points but with differing velocity profiles. A trajectory may speed up or slow down arbitrarily, but as long as its geometric shape remains the same, it is part of the same path equivalence class. Colloquially, similar to how we can think of an arbitrarily second order differential equation as a collection of trajectories (its integral curves), we can think of the geometry as a collection of tubes. Each tube represents a path and contains multiple trajectories (infinitely many of them), the collection of all trajectories following that path with differing speed profiles (see Figure 1).
Concretely, two trajectories are said to be equivalent, and hence along the same path, if they are a time reparameterization away from one another. Given a trajectory with time index denoted , a time reparameterization is a smooth, strictly monotonically increasing, nonlinear function denoted giving rise to a new time index . the time reparameterization creates a new trajectory . Since, for a given and corresponding , the points of the trajectories align
and this relationship is a bijection,333Formally, it is a diffeomorphism between coordinate charts of the same one-dimensional manifold of points. we can say that the two trajectories and follow the same path. Since and , we can see velocities and accelerations under the time reparameterization are linked to one another as
where notationally the dots are understood to be time derivatives w.r.t. their respective time indices. E.g. , and so forth. A smooth nonlinear geometry is defined by the collection of all time-reparameterization invariant paths in a space. For every point and speed-independent direction vector , the nonlinear geometry defines a unique path eminating from that point following the specified initial direction .
Denoting orthogonal projector projecting orthogonally to velocity as , the family of geometries we consider here are those characterized by a second-order differential equation of the form
where is a smooth function that is positively homogeneous of degree 2 (HD2) in velocities.444Generally, a function is said to be positively homogeneous of degree (abbreviated HD) if for . Homogeneity of degree 1 and 2 is used in the definitions below as well. In this case, must be HD2, meaning for . We call equations of this form geometric equations.
Since the projector is reduced rank, there is solution redundancy. We will see that this redundancy precisely describes the ability to arbitrarily speed up or slow down along a trajectory while sticking to the same path.
The interior equation in isolation
is called the generating equation and is said to generate the geometry via its system of trajectories. Note that since has null space spanned by , solutions to the geometric equation are solutions to
where is a smooth function defining an acceleration along the direction of motion. We call this equation the explicit form geometric equation.
All time reparameterizations of generating solutions are geometric solutions, and each geometric solution characterized by starting position and direction is a time reparameterization away from any generating solution with initial conditions for .
We first address reparameterization of generating solutions. Let be a generating solution trajectory and let be an arbitrary time reparameterization. In terms of its inverse (which always exists since is strictly monotonically increasing by definition), by Equations 2 we have
where . Thus, solves an explicit form geometric equation and is, therefore, a geometric solution.
Next, let be any solution to the geometric equation. Then at every point,
for some smooth function across the trajectory. Under a time reparameterization we get the following equation in terms of
Since is an ordinary second-order differential equation it has a unique solution for every initial condition . Under any of those solutions, the second term vanishes and we have (since ). Therefore, under such a time reparameterization, is a generating solution.
Moreover, since the initial condition defines the initial velocity which uniquely defines the generating solution moving in direction . Since any initial velocity can be generated this way, every time reparameterization of this sort maps to a corresponding generating solution by initial conditions and every generating solution can be created by such a time reparameterization. Therefore, there is a bijection between time reparameterizations solving and generating solutions with initial conditions . ∎
The proof of the above theorem shows that any time reparameterization solving
induces a generating solution. We are, therefore, free to choose the time reparameterization so that the generating solution’s velocity matches the geometric solution’s velocity at a given time (use the condition at the specific as the initial condition along with ). With that mapping from to corresponding generating solution whose velocity matches at , we can view the geometric solution as smoothly moving between generating solutions, using the redundant accelerations to do so by speeding up and slowing down along the direction of motion.
To illustrate path consistency, we design a geometry, , that naturally produces particle paths that avoid a circular object in coordinates, , as
where is a differentiable map that captures the distance to a circular object and is a scaling gain. More specifically, , where and are the circle’s center and radius, respectively. Furthermore, is a barrier potential function, , where is a scaling gain. Altogether, produces an increasing repulsive force as distance to the object decreases, and makes homogeneous of degree 2 in . For this experiment, , for two scenarios: 1) , and 2) for the initial conditions, . Eleven vertically spaced particles that follow the above geometry are initialized with the two different initial speeds. Traced paths at the two different speeds are overlayed as shown in Fig. 1. Noticeably, the paths generated are completely overlapping which confirms path consistency.
Iii Finsler geometry
Here, we derive a broad class of nonlinear geometries that arise from solutions to the Euler-Lagrange equation, known as Finsler geometries. We show that the generating equation also derives from the Euler-Lagrange equation, applied to an energy form of the geometric Lagrangian. These energy solutions are energy conserving and can thus be viewed as energy levels. Geometric solutions to Finsler geometries, therefore, smoothly transition between these energy levels (generating solutions) by speeding up and slowing down along the direction of motion while remaining along the same common path.
Iii-a The Euler-Lagrange equation
The Calculus of Variations  studies extremal trajectory problems. Given a function of position and velocity, known as a Lagrangian, and a class of smooth trajectories ranging between two end points and , we can ask which of those trajectories minimizes the “total Lagrangian” across the trajectory:
where is understood to vary per trajectory based on its natural time interval length. The integral is known as the Lagrangian’s action.
Figure 2 shows a simple (hypothetical) pictoral example. The dark trajectories depict possible candidates in and the dotted trajectory depicts the extremum. We won’t derive it here, but extremal solutions are characterized by solutions to the following boundary valued second-order differential equation:
This equation is important in its own right and is known as the Euler-Lagrange equation.555We write it in negated form here relative to the common expression from the Calculus of Variations  to match better with the equations of motion below, as we’ll see.
By the theory of ordinary differential equations, this equation (second-order) has unique initial-value solutions when it is pointwise well-formed for (we’ll see that this means its velocity Hessian is invertible for . This won’t be the case for Finsler structures (and we’ll get the solution redundancy characteristic fo geometric equations from that), but it will be the case for the corresponding energy form generating equation.) This uniqueness of solution means that for any , we can play forward the Euler-Lagrange equation to generate a solution trajectory starting from that position and veliocity. We can connect this solution back to the extremal problem by noting that every point encountered along this initial-value solution is a possible end point and the solution to the initial-valued problem solves the boundary value problem with boundary constraints and . In other words, every subtrajectory of the initial-value solution is an extremal solution between its end points. Importantly, this enables us to consider the Euler-Lagrange equation in isolation and understand all of its solutions as extremal solutions of the action.
Expanding the time derivative brings the Euler-Lagrange equation to a more concrete form and clarifies its role as a second-order dynamical system:
where plays a role analogous to a mass matrix and is a force-like object. Solutions to the Euler-Lagrange equation can be easily integrated forward from an initial position and velocity using the solved acceleration form . Note that this solved form is only well-defined when is invertible as noted earlier. While here we consider and as merely analogous to mass and force, we will see that this analogy takes on a deeper, more concrete, meaning under Finsler geometry, where we require the Lagrnagian to take on an intuitive form so that , for fixed , becomes a squared norm like measure of velcities, giving it an interpretation of “length squared”. Under these particular Lagrangians, plays a role of mass, plays a role of force, and the equation , or with is a geometry generator.
Iii-B Finsler structures
A geometric Lagrangian is a Lagrangian whose Euler-Lagrange equation can be expressed in the standard geometric form
where is homogeneous of degree 2 in velocity (HD2). Recall that such degree 2 homogeneity implies an invariance to time-reparameterization (see Section II) of the geometric solutions characteristic of a geometry (solutions are velocity independent paths, not individual trajectories). A Finsler structure is a particular geometric Lagrangian with the following nice properties:
with equality if and only if .
is positively homogeneous (HD1) in so that for .
is invertible when , where .
We call defined this way, the corresponding Finsler energy.
The positive homogeneity requirement means that the action functional is independent of time reparameterization. Recall that a time reparameterization is defined as with , so the action function has the property
This property, alone, suggests that solutions to the Euler-Lagrange equation of will be independent of time reparameterization. Theorem III.1 proves that conjecture. The speed independence of this action functional, in conjunction with the conditions of a Finsler structure listed above, also suggests we can view the Finsler structure as a generalized length element. The action functional is, therefore, a generalized arc-length integral.
Each Finsler structure has an associated energy form . Since is HD1, , so the Finsler energy is positively homogeneous of degree 2 (HD2). A useful property homogeneity is given by Euler’s theorem on homogeneous functions , which states that if is homogeneous of degree in , then
In the case of the energy Lagrangian , we have , so
The equations of motion under the Energy Lagrangian , which we call the energy equations, are
where and . Since these equations are known to conserve , and in this case , we see that this Finsler energy is conserved). is often referred to as the energy of the system, and is its energy tensor.
Note that the third requirement on Finsler structures given above is actually a requirement on the Finsler energy . It ensures that is invertable by definition when . The equations of motion , therefore, can always be solved to give a unique acceleration form
This equation is known as the geodesic equation, and we will see that it acts as a generator for the geometry expressed by the ’s equations of motion.
Without proving it here, we note that taking derivatives in reduces ’s homogeneity by 1 (a general property of homogeneous functions). Therefore, examining , we see
for which is HD2 already, and multiplies the HD1 making HD2 as well. So is HD2 in its entirety. Moreover, has two derivatives so it is HD0 (i.e. , meaning the energy tensor is independent of the scale of the velocity and, therefore, depends only on ’s directionality). That means is HD2, making the geodesic equation a generating equation, with associated geometric equation
The following theorem shows that this geometry is precisely that characterized by ’s equations of motion. We denote the geometric equations of motion by where and for Finsler structure . In this case, is reduced rank, as we will see, so we can’t solve for a unique acceleration. Instead, the redundancy is precisely that expressed by the geometric equation.
Let be a Finsler structure with energy form . Then the energy equations is a generating equation with the associated geometry given by the geometric equations .
We already observed that is homogeneous of degree 2, so is a generating equation where it is (uniquely) defined. Since is invertible by definition when , it is only undefined for . But for , solutions to are stationary point trajectories () since is homogeneous. Therefore, defining creates matching limiting behavior (independent of the characteristic properties of generating equations, which characterize geometrically consistent generating trajectories for ), so is a generating equation with solutions consistent with .
To show is a geometric equation, we calculate explicit expressions for and . , so . Therefore,
where we use and denote (this quantity is called the generalized momentum and has a recurring role in Finsler theory). We also use the following identity, which we will prove in Lemma III.2:
Denoting and noting (also proved in Lemma III.2), we see that is a reduced rank matrix with null space spanned by
Using a calculation analogous to that which we used for Equation III-B we get
so combining we get
We first show that (i.e. is orthogonal to ) and then use that insight to derive an explicit projected expression for :
From Lemma III.2 we also get , so the expression reduces to
again since by Lemma III.2. Therefore, .
Since is orthogonal to and where and , must be precisely the coefficient on needed to remove the component of along . Explicitly, must satisfy , which means
is another expression for . That means
Noting again that (see Lemma III.2), we have
is a reduced rank matrix analogous to , with null space spanned by .
Combining all of these expressions so far, we get geometric equations of omotion
However, and are related by the identity since
Therefore, the geometric equations of motion can be expressed
where is an orthogonal projector with null space (any matrix with spanning its null space would express the same equation). Solutions to this equation are exactly those for which lies along the null space , so they can be expressed as solutions to the unprojected equation
for any . Since by Lemma III.2, those solutions also solve