Traditionally, mechanical systems operate through the coordinated motions of their rigid parts, joined together by joints such as hinges, sliders, and pivots. While these designs can be highly sophisticated, they quickly become complex as the number of parts increases, making them more expensive and time-consuming to produce. On top of this, rigid parts will inevitably degrade over time through friction and wear, and often need to be replaced.

In contrast, ‘compliant’ mechanisms are made up of flexible parts, which transmit motion to each other via their own elastic deformation. This flexible behaviour allows these systems to seamlessly integrate structure with function: reducing friction, complexity, and the overall number of components, making them far easier to build and maintain. Thanks to these advantages, applications of compliant mechanisms are now being explored in fields as wide-ranging as aerospace, medicine, and robotics.

Despite their promise, compliant mechanisms introduce a whole new set of challenges. When designing mechanical systems, engineers must be able to model how the system responds to applied forces. Unlike more traditional ‘dynamic’ designs, which focus on the forces a system will generate and experience, compliant mechanisms are ‘kinematic’ designs, which focus on how the system itself will move and respond to these applied forces. But unlike rigid mechanisms, which often follow straightforward equations of motion, compliant systems tend to behave nonlinearly – meaning their outputs don’t scale proportionally with the forces or inputs applied.

In past studies, researchers have attempted to capture this complex behaviour using a range of computational tools – including optimisation algorithms and models that represent mechanisms as networks of modular building blocks. However, one critical factor has remained difficult to incorporate: uncertainty.

In any mechanical system, there will always be some uncertainty in key design variables. In rigid systems, this uncertainty is usually easy to track – small changes in input tend to cause small, predictable changes in output. But in nonlinear systems such as compliant mechanisms, uncertainty in design parameters can be significantly amplified, making the system’s behaviour much harder to predict. As a result, current approaches to designing compliant mechanisms have become increasingly complex and computationally demanding.

In their latest study, Ahmed Alhindi and Dr. Meng-Sang Chew developed a new method for designing compliant mechanisms based on dimensional synthesis. Through an approach named ‘fuzzy logic’, they introduced a modelling framework that accounts for uncertainty in the system’s design variables – producing designs that are both more robust and computationally efficient. Their results offer a promising alternative to existing techniques, potentially making it far easier for engineers to create compliant mechanisms tailored to specific applications.

At the heart of Alhindi and Chew’s study is the idea that uncertainty should be addressed early in the design process. Traditionally, engineers assume that all design parameters are known exactly – but in real-world environments, this level of precision is rarely achievable. Variations in loads, material properties, and manufacturing tolerances can all affect how a compliant mechanism behaves.

The researchers proposed that by building these uncertainties directly into the design framework, it should be possible to create mechanisms that are more adaptable and reliable in practice. To put this into action, they applied a technique called dimensional synthesis. Instead of starting with a design and fine-tuning its parameters, this method starts with a desired output – such as a specific motion – and works backwards to find the optimal dimensions and parameters needed to produce it.

To model uncertainty, Alhindi and Chew turned to fuzzy logic: a mathematical tool that represents variables not as fixed values but as ranges, each with an associated degree of likelihood. These ranges, known as fuzzy numbers, allow engineers to work with imprecise data while still producing meaningful results.

When applied to rigid mechanisms, fuzzy logic can account for uncertainties that accumulate over time as the system degrades – for example, due to wear and tear on joints and materials. In compliant mechanisms, by contrast, it helps engineers model the wide range of possible motions that can arise from the flexibility of the system’s components.

To build their framework, the researchers adapted the Freudenstein equation – a tool widely used in engineering to relate inputs and outputs in mechanical systems. They reformulated the equation so that it could accept fuzzy numbers as inputs, capturing a full range of possible design scenarios. To solve the resulting equations, they used Newton’s method, a well-established algorithm for solving nonlinear equations. By adapting this method to handle fuzzy values, they ensured that their solutions remained stable and accurate even under uncertainty.

The next step was to refine these solutions using fuzzy optimisation. Since the initial solutions to fuzzy equations often overestimate the range of possible outcomes, they can often make the system seem more uncertain than it actually is. Fuzzy optimisation helps to narrow down this uncertainty by finding the best possible solution within a fuzzy environment, which balances performance with robustness. In other words, it ensures the final design performs well not just in ideal conditions, but across a range of real-world variations.

To demonstrate their approach, Alhindi and Chew applied it to a classic function-generation problem: designing a compliant mechanism to follow a specific motion path. Using fuzzy logic, they accounted for uncertainty in the input parameters and then applied their synthesis method to design a mechanism that could achieve the desired motion. The results showed that their design could closely reproduce the target output – even when subject to substantial variation in its inputs. Altogether, this demonstrated both accuracy and resilience in the final mechanism.

Alhindi and Chew’s results mark an important step toward more resilient and adaptable designs of compliant mechanisms. By embedding fuzzy logic into the dimensional synthesis process, they’ve shown that uncertainty can be accounted for directly, without the need for overly complex or computationally expensive methods.

As the demand grows for lightweight, flexible, and precision-engineered systems across a wide range of sectors, their work lays the foundation for a new generation of compliant mechanisms – designed not just for ideal conditions, but for the messy, uncertain reality of real-world engineering.