Written by Louis Fernandez, PhD Researcher, based at UTS and part of the Human-Robot Interaction program.
Mathematical optimisation often comes across as intimidating because it’s wrapped in abstract mathematical symbols and notation. If we peel back the equations, what’s the mathematics actually doing in a physical sense? Let’s build an intuitive mental model by looking at how we calculate the minimum distance between two shapes. This problem can be mathematically written as follows:
When you break down an optimisation problem, it comes down to two main components. The first component is the objective function, which simply defines our goal. In our example, the objective function is written as minimise ∥xr − xO∥2. In everyday language, this simply means our goal is to minimise the distance between two points, xr and xO, by adjusting where those points are placed. The expression ∥xr − xO∥2 is the mathematical language for the distance between these two points.
- Figure 1.1: Imagine two random points in space that can move freely in any direction. In this scenario, the closest they can get is zero distance (i.e. just put them in the same spot!)
Imagine these two points floating freely anywhere in space (see Fig. 1.1). You’re allowed to move them around however you like to make the distance between them as small as possible. What’s the absolute shortest distance you can get? It’s zero, because you can simply place both points at the exact same location. On its own, this result is trivial, which is why we need rules to make the problem useful.
This brings us to the second component, which are the constraints of the optimisation problem. Constraints set the boundaries and limit how our points move. In our equation, the constraints F (xr) ≤ 0 and F (xO) ≤ 0 act as inside-outside functions.
- Figure 1.2: The constraints state the the points xr and xO must stay within their respective shapes. In this case, xr must stay within the blue region and xO must stay within the green region.
An inside-outside function F tells you where a point sits relative to a shape. If the point lies outside, F returns a positive value. If it lies inside or right on the surface, F returns a negative value or zero. Because our expression specifies that F ≤ 0, this rule forces our points to stay inside or on the boundary of the shape. Instead of letting our points float freely anywhere in space, these constraints tie them to real physical objects, such as a robot arm and an obstacle in the environment. We strictly require that the points xr and xO stay trapped inside or on the surfaces of those shapes (see Fig. 1.2).
- Figure 1.3: The optimisation problem gives us the minimum distance between two shapes as the objective function acts as an attractive force the pulls xr and xO toward each other but the constraints limit how far they can move. This ultimately gives us the minimum distance between these two objects.
Now, let’s combine the objective function and the constraints to see the full physical picture. The objective function acts like an attractive force pulling the two points toward each other. At the same time, the constraints act like physical barriers that stop the points from travelling outside the regions defined by the shapes. As the optimisation problem is solved, the objective function pulls the points as close as possible until the constraints stop them at the surface. What the solver ultimately finds is the minimum distance between the two objects (see Fig. 1.3).
1.1 Why Optimisation Matters in Robotics
Hopefully, this mental model gives you a clearer intuition for how mathematical optimisation works in practice. This framework is essential in modern robotics, where calculating distance is rarely as straightforward as applying a basic geometric formula. While simple shapes like spheres have exact analytical formulas for distance, real-world objects require far more complex geometry to be accurately modelled. A robotic arm with multiple joints or a human worker cannot be simply captured by using spheres and ellipsoids. In my PhD research, for instance, I used shapes called superquadrics to represent complex robot links (see Fig 1.4). Mathematical optimisation provides an efficient means of evaluating distances between various robot–obstacle pairs, where the bodies are modelled as superquadrics.
- Figure 1.4: By representing robot bodies (blue) and various obstacles (red) with more com-plex geometric shapes, we are able to more accurately capture their geometry. From here, we can use mathematical optimisation to calculate distances between them in real time. Fast, reliable distance computation makes robotic systems far safer and more efficient, which is a crucial requirement when robots share unpredictable spaces with humans.
By framing minimum distance checks as an optimisation problem, robots can determine how much space they need to remain safe in real time. For example, a factory robot working alongside people may need to recalculate its distance from nearby workers and obstacles dozens of times per second. By continuously solving this optimisation problem, the robot can monitor its surroundings and respond as conditions change. It can adjust its trajectory, slow down, or stop completely when necessary to prevent a collision. This allows robots to move safely and autonomously in unstructured, human-shared environments.
1.2 Optimisation in Daily Life
While robotics relies heavily on these principles, you actually interact with mathematical optimisation every single day without realising it.
- GPS Navigation: When you ask your smartphone for directions, an optimisation solver can help find the best route to your destination. The objective function might minimise travel time or the number of toll roads used, while the constraints include the available road network, speed limits, one-way streets, and current traffic conditions.
- Smart Home Thermostats: A smart heating controller balances comfort against energy costs. The objective function minimises electricity usage, while the constraints ensure the room temperature stays within your preferred comfort zone despite the weather outside.
- Delivery and Logistics: Courier services must determine the most efficient sequence for a driver to visit dozens of delivery locations. The objective function minimises fuel consumption and distance driven, constrained by delivery time windows and vehicle cargo capacity.
Once you view the world through this lens, optimisation stops looking like intimidating equations on a whiteboard. It becomes a practical tool for finding the best possible solution while respecting the physical rules of the world around us.